Every system has had to wrestle with the 'fighters are mundane, wizards are special' meme. 4e just handled it better than any other D&D has. Having played 4e wizards and 5e wizards I don't find the 4e ones less wizardly or less fun, just less endowed with an overabundance of 'stuff'.
By 'stuff', you mean spells? Yes, wizards should have spells. Lots of them. And be able cast more than 3 per day.
A wizard in 4th has a total of 3 dailies. To call that a nerf compared to a pre-4e wizard would be a huge understatement.
You reach level 16 then finally you can fly, for 5 minutes, once per day max. Wow. How exciting? A level 16 wizard in 5e can fly for hours a day, if that's what he wants to do. It's versatile, how you spend your spell slots and on what. It doesn't presuppose that you're going to use them for combat. Sometimes utility spells are super useful in combat. A high level 2e wizard can effectively fly all day long if he wants to, or make his whole party fly to get to the top of the castle. In 5e you can't do that because of concentration, but you can still at least explore yourself, or let the fighter or rogue sneak ahead.
Same thing with a fireball. You can never cast more than one in a day in 4th edition. Why not? Not because of balance, surely. Why can't you use your level 9 slot to cast a level 5 spell a second time? I was needlessly punitive and rather stiff, beyond any balance considerations (the other level 9 spells would be probably superior, though not always). There were times when I would level up and my older level dailies or encounters were strictly better than the new options that just became available. How is that balanced? Higher level powers should be more powerful, obviously. Except they weren't always.
I agree that 5e doesn't go mad with caster power, but when you have some people complaining that fighters are so powerful and damaging that it's useless to play an evoker just to catch up to a fighter's peak-encounter-damage levels a couple times per day, and in other places you have people saying that wizards totally mop the floor with fighters after level 5! (that is a fantasy).
Compare the fly spell of 2nd edition, 4th, and 5th. The power of those versions of Fly seems to me to be like 2e >>>> 5e >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 4e.
4e magic took a back seat to class balance. And suffered for it. Hence, why we are playing 5th edition. Which made some fixes but didn't do it by making magic and magic using classes so mundane and weak that the Martial Power source was both magical and stronger.
Compare a level 5 wizard in any edition, to a 4th ed wizard of any level. The 4th ed wizard can barely do what earlier wizards could, and so ineffectively and constrained that it's supremely frustrating to even try. He certainly can't use his Fly spell for exploration. 5 minute encounter-sustain systemic limitation prevents that. Invisibility used to last 24 hours in 2e. Now it's much more reasonable. But it's not 5 minutes max, once per day. That is a whole other set of chains. Those chains exist because the entire game was designed around the combat encounter, and every single power and duration was designed to fit into that blinker tunnel vision mentality, and that's why it was fundamentally impossible to try and play a 4e game and imagine that a wizard was a powerful class with powerful magic. It wasn't. It was weak, limited, chained up and thrown into a dungeon without so much as a torch.