I'm not 100% sure I agree with you. The burden of proof is on someone saying that Sea Lion is feasible, because the default situation is no invasion taking place. So, once someone says, "It was impossible for the Luftwaffe to attain the required air superiority," as far as I'm concerned, the burden is on the pro-Sea Lion camp to illustrate how to invade the British Isles, not on the opposition to demonstrate any of a hundred other ways Sea Lion can be stopped.
That being said, I think the original plan the Germans had is an important touchstone. When I think of Sea Lion, I consider that some of the basics of Fuhrer Directive 16 are in play, because some of those basics are so fundamental that they must be addressed. Say what you want about barges, Bismark, paratroopers, or Stukas, but some parts of the plan are pretty damn fundamental no matter who is doing the invasion.
Directive 16 clearly states that the Luftwaffe must control the sky. There is no other option to be considered in this case; there is no substitute. Hell, even with air supremacy, the Allies had anti-Stuka blimps covering the invasion beaches to prevent the slightest chance of bombers messing with the assets off the coast. What other options does Germany have? Is there some panacea I am not aware of that could get Germany around the air power problem?
Directivee 16 also indicates that the RN needs to be out of the way. It doesn't have to be sunk, although that is one way to achieve this goal. But however you do it, the RN has to be out of the way. There is no other option to be considered in this case; there is no substitute for "holy crap, the RN is shooting at our transports!" Even a handful of capital ships could prevent an invasion or make it messy until they are sunk. Now, I have mentioned in this very thread my idea of thinning out Britain's capital ship numbers in cooperation with the IJN and RM, but even then, the goal wasn't "We can cross the Channel while the RN has ships left" but "Britain either has to give up 100% of her overseas possessions and lose contact with Dominions so she can prevent Sea Lion, or try to hold on to the overseas possessions while remaining insufficiently protected at home."
To me, arguing over barges, Luftwaffe expertise at sinking ships, amphibious capabilities of German forces, naval mines, and shore batteries is interesting, but none of it changes the two fundamental problems outlined by Directive 16. I also happen to think that if Germany some how overcame the two problems above, it might also be in a position to solve the smaller issues with barges and whatnot. After all, give me from 1936 to 1940, I can give you enough transports to cross the Channel in style, and train the Luftwaffe to sink ships reliably, and have better naval mines and mine sweeping, and better utilization of shore batteries. It's a pittance compared to building a navy that can secure the Channel or an air force that can defeat RAF Fighter Command.
One thing is certain about the germans, they were just as ingenious as the allies. I am in no way pro-axis but i do feel they are consistantly underestimated by historians because of one factor. Hitler. They never mention him as the reason but he is behind every single bizarre early success while also behind every failure once the allies started playing ball.
So lets look at some of your points. Before i start, ill make it a point to state that i believe sea lion would have been impossible at worst and extremely difficult at best.
For devils advocate sake, lets play a very detailed arm chair historian role to see what could have happened. I am doing this for fun so just bare with me. It will probably be long.
1. Air supremacy. Almost every historian agrees the luftwaffe literally had the RAF to such a point that within a matter of days, if the assault had continued, air defense capabilites of the isles would have been severely limited in effectiveness. How we translate this can be various but the end result begs the question of how much further could the luftwaffe push home the attack and what would the combat effectiveness of the RAF be during an invasion? What changed? Well we all know Hitler altered the strategic goal of the battle of britain to terror. This gave the fighter defenses breathing room and now we are back to square one. The invasion is impossible. What if the strategic goal had not shifted and appropriate commitment of war production and training to air combat capability been given to the luftwaffe? This could have led to the initial requirements being met. Hitler of course could not have been involved. Once again, as we all know, his ideas early on were difficult to predict therefore offered surprise/shock value and produced results. Once he got involved during the events, he screwed them up quite well generally. Especially mid to late war.
2. Sea power. Regardless if the U.K. decided to abandon its colonial empire to fully defend the home islands(which i doubt it would) the germans would need a full proof plan to allow for transport of troops, reinforcements and most importantly supply. For arguments sake we will assume the germans have air supremacy. What did the germans have at their disposal and how much time would they need to get two army corps across the channel and supply them? Well they had, as you stated captured french coastal batteries and batteries that they were constructing. Some of these batteries were massive but were not built till after the cancelation of further battleship production and cancelation of sea lion. What if dunkirk were pushed home fully with the annhilation of the entire british expeditionary force, the operation halted till the todt and other batteries could be built and proper quantites of powerful anti ship mines produced for sea corridor establishment? This could have led to a drastic reduction in home guard defense capabilites allowing for a delay of when the operation were to be conducted, allowed for massive and devastating channel firepower and opened up production for a multitude of other options to help push home an invasion attempt along with supply cutting options. Sub production and so forth. Most historians agree if anti shipping warfare had started earlier, it would have been even more devastating. Once again, hitler with his bigger is better mentality screwed up the german navy with grandiose concepts of a navy consisting of BB's far larger than even the Bismark which, if memory serves, was actually 3rd in line to the largest of the classes of BB's he had in mind. So, overwhelming coastal batteries mixed with effective mining of the channel, air supremacy and a direct focus on submarine and further air production puts us in at least a position to assume there was at best a minimal chance.
3. Invasion fleet. Anything is possible. A joke right? Well lets use the reverse to show just how true saying anything is possible is. At dunkirk, the british evacuated their entire expeditionary force along with some scattered french forces in about a week. I am trying to remeber the total number but it was something north of 300k troops. No immediate organisation options. No known way to do it. This isnt two corps of troops we are talking about. This is army and army group size we are talking about here. All this during a war and during pressured combat where anything is possible. So lets look at it from a perspective of the germans. They have a window where they can, with reasonable air protection, get troops across. The channel is heavily mined compensating for the lack of naval capability and further backed by exceptional coastal battery support. How do they get across? It seems almost comedic to have an army cross in tug boats and fishing trawlers doesn't it? Well i agree. At best id save that kind of passage for supplies if anything. I just can't see that happening but anything is possible. At this point id have to suggest further commitment to proper landing craft which COULD be possible if the proper commitment was made without interference from Hitler. Again, another uncertainty.
4. Supply. The single most important factor. Getting it, keeping it and making it effective. I have read studies that did hard mathmatical crunching of numbers to show precisely how much supply a german rifle, mechanized and panzer division would need to remain in a full combat effective state upon landing on the isle. These numbers, while in the hundreds of tons, are lost in my memory. I wanna say it was something like 600-800 tons a day or something like that. Thats combat supply and most probably for a Pz div. You can correct me if i am wrong about that. Regardless, the point here is supply shipment. The study pointed out what a reasonable port to landing site shipment tonnage could have been and also a french fully operational port to british fully operational port shipment could be in tons. Two things are taken into account. The french port is fully operational and so is the british port. The study showed that the germans would have a net loss of supply shipment the first week of invasion at a nonport landing site severely depleting the combat capability of the divisions landed and only making it worse as time progressed as more troops landed. Upon capture of a british port, assuming the brits didnt completely annhilate the port, the study showed that even the best southeastern port in fully operational order could only have received upwards of 800-1000 tons of shipping per day. The only option the germans had would be alternative shipment methods. Planes being the most probable but even here we have a problem. During stalingrad, there was an attempt at air supply and this was horribly inefficient and insufficient.
So to summarize. I cannot get passed the supply option. They might have been able to get air superiority. They might have been able to control the channel just long enough with very specific options used. They even might have been able to get across the channel and land making some headway into the country. The problems i run into once they land are ammo, dietary consumables, maintenance capability and other misc supply needs. Put simply, they would have starved or turned into foraging battlegroups further disrupted by lack of fighting equipment needed to remain combat effective causing morale to drop.
Just thought id have fun with this. Never really wrote down my thoughts on it before.