My sources of perplexity with Fate Core (I would not call them “objections”) are two, one much more substantial than the other. The substantial problem is how heavily “meta” the game must constantly go. In traditional games, for players to talk not about what their characters were about to do, but about the kind of character they were, the in-game abilities that they had etc. was the exception rather than the rule and, at least in many people’s opinion, was in fact the low point of a session. Most people worked from the assumption that the core of the game was acting the characters, not talking about them. Fate demands that the implications of the aspects for a character be discussed by the players and the GM at every turn: just read the game examples in the book. “Hey GM, don’t you think that since I am The Most Famous Ventriloquist this guy should know my name?” “Hey Player, don’t you think that your Anger Management Problems should kick in now that the guy is calling you a sucker?”. It is not the exception, it is the rule. This, besides feeling “artificial” to me, also runs the risk of changing games of Fate into “tvtropes: the role-playing game”. In practice, players are encouraged to resort to highly recognizable features that can be agreed upon by other players and the GM without too much hassle, in other words: resort to clichés. The aspects themselves, in the examples given in the book, are phrased in terms of fictional clichés. This might be regarded as a weakness: while traditional RPGs built quite a lot on this or that cliché, they were not themselves built upon clichés.
The second source of perplexity is that you can abstract reality away only up to a point: sometimes, it is detrimental even in a perspective of fiction emulation. Let us take the most stereotypical (but by no means unique) example: damage. The default assumption in Fate Core is that every weapon (including your bare fist) inflicts the same damage and armour is only there for show. You are given options to introduce a finer grained treatment of arms and armour, but the authors discourage you from doing so with an argument which sounds somewhat disingenuous. What is the point, they say: if you say some weapons make more damage, the PCs will want to have them, and you will have to give them to the NPCs in order to keep some balance, so you see, it’s all zero-sum. Well, not really. The existence of a range of different damage classes for weapons and of protection values for armour opens interesting tactical problems, not to mention the fact that it is generally assumed even in a non-simulationist, purely fictional perspective: in few (if any) movies or novels you take off a tank by throwing stones, but in Fate nothing (nothing mechanical, that is) prevents a PC from succeeding at that. Mind you: one could argue that the default view fits nicely in the general architecture of the system. More precisely, of a system that has a greatly abstract take on injuries and death. In Fate whenever a PC takes damage the GM and the player simply decide what consequences the PC must take, with only very general guidelines to follow. Death for purely technical reasons (“the PC took too much damage and died”) is virtually absent from the game. PCs should die only insofar as it is cool for them to do so! Also, PCs and NPCs are dealt with altogether differently. Average NPCs (“mooks”) cannot even take consequences: if they are hit, they are taken out. In a way, abstracting away from a lot of technicalities makes sense in the context. Still, all this is a far cry from everything we knew from traditional games: not everybody wants this.
So, in sum, would I play Fate? Of course. But I would only go for it depending on the players and the setting. Very cinematic adventure settings (think Indiana Jones or Star Wars) would work especially well, other stuff less so. As for players, I am under the impression that Fate could only really work with very experienced players, better still if they have quite a bit of experience with GMing as well. It presupposes a high degree of awareness of the problems of writing settings and scenarios, not to mention a degree of ingenuity (especially in exploiting aspects) that many players (those who have never GMed or have but hated it) might just not have under their belt.