My critique of modern Fire Emblem:
1. Lower access to characters - assuming you try to play classic mode if you lose a character, chances are close to certain you lose access to getting characters later. In the older games, from a gameplay perspective, you are usually good to lose a lot of characters because you get a sufficient number of new ones to replace the dead ones. This hasn't been the case in Awakening or Fates (not sure about Valentia, Echoes, or Houses as I haven't played those).
2. Games not balanced for classic mode - You may as well call it "no fun mode" - as I pointed out in my last point, if you lose a character, it's fairly devastating. But there are other reasons, there are a lot of "Got you!" type maps where it's not even a case of "I have to sacrifice a unit to win this battle," and more of a "SIX OF US SPAWNED BEHIND YOUR LINES AND WE'RE ATTACKING NOW AND KILLING YOUR HEALERS AND ARCHERS!" - I noticed this particularly in the Birthright game of Fates. This is fine if you are playing casual mode, but it's pretty much game over if playing classical mode. It adds a new dimension of memorizing enemy spawn patterns, which is not fun at all since they are not logical as they were in old games; and this is why I strongly feel they didn't try to balance or playtest much around classic mode.
3. The story focus is long gone. While there was clearly a decline in story quality (with the exception of maybe Path of Radiance which was one of the few games who had a better story than its more immediate predecessors) the games became much more generic feeling, the relationships felt more like placeholder text and less like a real relationship that you see developing. Additionally, the focus on narrative has been lowered significantly. The stories are less epic, less mature, and more "anime" for lack of a better term. But this is a reasonably general thing that has happened in strategy RPGs in general: Look at the stories and epic feeling of games like Fire Emblem Genealogy of the Holy War, Ogre Battle, Final Fantasy Tactics compared to Disgaea 5, Fire Emblem Awakening, and Final Fantasy Tactics Advance games; it's like when an R-rated film has a rated PG sequel. I don't want to sound like I am bashing those games, I LIKE those games, but they are not the same as the strategy RPGs of old.
4. Simplified gameplay - arguably a good thing, but not for me! I enjoyed the older games where you had to more carefully consider your moves, do a lot more calculating, and take a risk, while in newer Fire Emblem there are fewer variables to consider (except random spawning); but ultimately it makes victory taste less sweet. Not once since Awakening has come out do I get that "Holy hell I can't believe that worked!" feeling. And I'm not even talking about the elemental stuff, but rather how the game presents information to you.
5. No deaths = no meaning - in the past games, while from a gameplay perspective losing a character wouldn't screw you... it would be more like losing 12-20 characters that would. However, the sense of loss was immense, and it would change the story of the game. It added to the game's replayability; each experience was different than the last... As well in Blazing Blade, you could go through with the Hector perspective instead of Eliwood for a more rowdy point of view (and also a little more challenging), Sacred Stones allowed similar branching.; anyway! I digress. Fire Emblem Awakening is a game you might play once, or twice again to see what a harder difficulty/mode is like, but generally you wouldn't play through the same game 8-12 times no matter how big of a fan you were because the game focuses more on character developmental chores than it does on a straightforward epic strategy-packed adventure; to me, that's all the fun stuff and none of the chores/filler. Anyway, because Awakening and later games are balanced for casual mode, the other mode (with deaths) is not very fun at all, and so it's in your best interest not to use it; but it's a catch 22 because playing casual essentially turns off the tension/drama/sense of loss that the older games provided.
So, for me, going back and playing on those really out of date graphics and interface is still more fun than the much more advanced and more beautiful looking recent games (I am unsure how I feel about the look of the new game just yet).
That's just me, and I definitely fall more into the fan of the classic Fire Emblem games fanbase, the first one I ever played was Fire Emblem 4 on emulator; although I would say roughly 50% of all Fire Emblem I have ever played has been specifically FE; Blazing Blade (Fire Emblem Advance, the first one officially released in European markets).