Is there a downside to this strategy of using spies to get claims on you neighbors' provinces?
No... not unless you have other more useful CB to use that can make you grab more or with less AE... I always makes claim especially early game and also late game since it saves admin points...
The difference is marginal late game since you only get a 10% to coring, meaning if you are coring a territories you lose the 10% on the last 50% coring cost to make it a state... For shit dev provinces I tend to not claim unless I have a truce and the spy network to do so.
Also note that it takes less time to core when you have a claim, which actually have an effect on corruption and income since OE increases corruption (price of monarch point increase and less income due to paying it down) and trade hurts a bunch... Also there are some other modifiers associated with OE, like merc cost increase (less income), dip rep (usually not an issue but can be in edge cases), unrest (duh) and improve relations... Last one is used for AE decay so having that modifier as short as possible (it´s a yearly modifier, so should be okay unless you pass 1 jan) is a good thing if you want to limit coalitions a bit...
Oh... and the 10% autonomy reduction... given... only useful when you need to state it, but have an impact on income, trade, manpower, sailors you get from it... The claim, if you state it, is equal to 10% benefit you lose from the province which in high dev provinces can amount to a bit... And since, early game at least, autonomy usually only goes down at peace with the pace of 0.10% monthly it will take 100 months for you to get the same income and manpower from that province than if you had the claim...
Also... worth noting, that keeping a high spy network gives you (100) +20 siege ability that translates to shorter siege times and faster wars, especially useful those pesky mountain forts... And -10% AE in that nation... so if you have a nation nearby you don´t want to be affected as much with AE keep the spy there... else if you keep it in the nation you are at war with... should also limit the impact on AE from him, since the target nation always gets the highest amount of AE (duh).
It's slow. Also the Conquest CB doesn't give any reductions to aggressive expansion.
It´s slow... sure... but I have little trouble getting claims on the provinces I conquer... Early game I focus on humiliate, war rep and gold since AE is higher there and you kinda need PP over 50 as fast and long as possible, you need MP to dev, core and get ideas not to mention that early game is where it´s best to get the mil tech difference and unlock idea groups and such...
You cannot get enough claims and you will pay dip points for any extra province you take (those are "unjustified demands"), so people tend to use other CBs later in the game
You can... Or else I´m imaging how I play this game... there is benefit though by having claims which are not as apparent for most... not just the dip cost...