I loved the original X-COM. You're right that it felt so much bigger and full of options, there was so much more you could do in it than in the remake (which wasn't bad, but I didn't feel the need to play-through again after finishing it).
The battles in X-COM were another example of this: the maps felt bigger, you felt like you had less idea where the enemy was likely to be, since you had more soldiers you felt like you could afford loses, you could be ambushed from more directions and the game really made you pay if you let one of your guys go somewhere unsupported. It also sprung "Oh F%^&!" moments on you more - the time when a whole squad of troops bumped up against a single Muton armed only with laser rifles, ordinary grenades, and ordinary launchers (not the big heavy launchers which might actually take one down): it was like something out of Predator, with my guys just unloading on this single Muton with everything they had only to find that they didn't even leave a scratch. Eventually one of them managed to get close enough to use a stun-stick, but only after half of my guys were killed/wounded.
The characters in the game also developed more organically - they weren't rail-roaded into a specific speciality with hard-coded limitations, but instead could be jacks of all trades. The fact that there were more of them meant you felt less of a compulsion to save-scum every time someone got hit.
EDIT: I often felt the levels in the X-Com remake looked like they had been pre-set and weren't procedurally generated. Having checked the Wiki page I now see that this was true.
The new Xcom 1 tactical battle maps were very finite in variety, to the point where I memorized most of their tactical layouts in one play through. And I was playing on slightly hard. Well, part of the reason why old Xcom 1 players don't find a lot of replayability in new Xcom 1 is that their strategy and tech is almost exactly the same... which basically means if you remember your old Xcom 1 strategies, you could basically start playing on hard or what not and go exactly into the tactical configuration you need to fight escalating alien threats. The fact that the research, production, and upgrades of soldiers and alien tech got easier, just means you didn't get surprised by the difference in game mechanics.
The mutons were definitely hardy, and the stun whatevers came in handy, since they produced fatigue faster than some weapons ate up hp. I seem to recall there was a ranged weapon stun... oh yeah, smoke. I just threw demolition charges in 3 point ambushes, and smoked those mutons out, while frog footing backwards with my squad. It was a pain in the a carrying around those huge charges though, but very useful for breaching house walls and sometimes even alien alloy hulls (if you weakened it with some plasma fire first). Hard to remember, but it was the snake peeps that I fought with lasers, and that was similar to your muton fight. And the biggest scare was the damn saucer tanks in terror missions. Their accuracy was not normal... and the sound of their engines freaked me out, and the first time I encountered them was at night, so I couldn't even see them properly.
Next after that would definitely be the chrysalids.
Another funny thing is psionics. That was the big "secret tech" upgrade, that nobody knew about, except old xcom 1 users. That changed the tactical situational immensely, especially since I was getting mind controlled in my sky ranger in old Xcom 1 and did not realize where it was coming from. I was blowing up half the alien ship trying to stop it too. And the thing is... psionic potential was completely random back then, and I had no idea who was strong or weak, except for those guys that kept getting mind controlled.
There were also situations where the sky ranger or the upgraded ones landed, and the aliens surrounded me with an alien launcher, basically mini cruise missile with multiple stage rockets to change trajectories and using some kind of super explosive, or anti matter, blowing people up. I was always scared of being trapped in that gang plank like Saving Private Ryan at a beach. One time I put all 12 of my super commandos at the bottom of a large alien battleship lift, thinking I would go straight up to the command center.... I took so long doing so, that the alien up top launched a missile below and killed almost everyone in powered armor. I was like.... "maybe I was being too cautious there". He "heard" enemies down below and must have seen a single glimpse. Next time I was in that situation, I decided to breach the command room externally, from outside the ship, using my own alien launchers: then fly up to the level with jet packs. I was trying to save elerium and alien ammo though... last time I tried that I think my charges blew into the ship and blew up some valuable stuff.
Xcom 1 old soldiers were definitely more layered and unique. The skill rpg system in the new Xcom was designed for people used to a simpler, more abstract system. Crusader Kings 2 simulates dynasties using a variety of traits and skills, some are good others are horrible. And in the old Xcom 1, because you could just keep hiring soldiers depending on time or money, what you got was this clump of clay that you would mold over time. Because the new Xcom 1 allowed you to directly influence the soldier's perks and skill trees, they didn't feel simulated characters. More like weapons in and of themselves. The roles they did have, were pretty cool and felt really nice to play, like sniper. It just didn't take as much work. In Ck2, you need to painstakingly educate your future heirs, if you want the edge. That attention to detail, that work, separates out what humans place value in.
Because they were pre-set.
On a side note, i liked the new Xcom more than old one. True, it have lot of faults, like the useless and costly gene enchancement as opposite to overpowered mech troopers, the obnoxious satellite system and functional cut-off of the air combat (bacause what is there is placeholder at best), but the few things also totally won me out - lack of multiple bases is just god-send, i hated this with passion in every x-com version and clone, also the lack of chasing last alien in some weird places in combat. About the TU's i'm rather undecided. They were functional, but the realtime of Ufo:whatever series worked good too and the remake x-com 2 action points is good too. If anything, it make the combat faster and no less tactical (at least until you are able to steamroll everything, but old X-com had that moment too).
I figured out early in old Xcom 1, that I only needed bases to intercept and shoot down alien ships. The actual landing site wouldn't disappear for a day or so. So my sky rangers and important facilities were kept at the primary HQ, with everything stockpiled in one central location, making it easier to defend against alien incursions. But the outlying bases only had to fit hangers and supplies for them. This reduces the micromanagement. People who tried to put soldiers or workers at two bases... that was probably not a good idea. I only ever needed 1 primary, 1 secondary, and 1-2 tertiary fighter bases. The secondary was my second squad, but came very late in the game.
Xcom Apocalypse, the third one, used real time for tactics, but the previous 2 were always turn based, but real time on the strategy map.
Xcom 2, Terror from the Deeps, was almost like a total mod of Xcom 1. The game mechanics didn't change much, but the background and new enemies were so interesting I didn't pay much attention to the popular criticisms. And I loved my melee weapons, it was so risky getting close to those enemies like the crabs, but so satisfying to cut them down in one or two shots, while ranged weapons were taking forever. 5+ shots.
On the topic of Moo2, it mostly reminded me of Master of Magic. I played MoM and Moo1 at about the same time, but they probably overlapped. I looked at the gameplay of Master of Magic from my current pov, and while the tactical battle is just as good as I remembered it, the OCD clicking on turn button isn't something I can tolerate easily now a days. Probably why I prefer real time grand strategy games now.
The chasing down enemies, especially in tight CQB situations like ships or cities in old Xcom, was pretty difficult. Since one had to maintain combat formations or else "surprises" may happen like an ambush. I started adapting and changing my combat formation into a broad sphere, and sometimes that single alien gray survivor would sneak around behind my search perimeter. So when I went back to check my skyranger, he shot my guy running out of TUs in the head. At day time, this wasn't so bad. At night time in farms without flares, it could get unnerving.