I felt this a lost opportunity for 5E. During the playtest, they kept talking about how level 1-2 was the apprentice tier. This gave the impression that most characters will start at level 3, unless the DM wanted to run the apprentice tier. If this would have been the case, they could have fixed much of multi-classing by spreading the abilities (such as proficiencies) across all 3 assumed starting levels, so that the 1 level dip wasn't as amazing as it is now.
The only feats I alter at my table are GWM/Sharpshooter. Everything runs as-is and hasn't had any issues for our game play. My only change to those two is a pretty common switch from -5/+10 to -Prof. Bonus/+2x Prof. Bonus just to get a little of the "boom" and swinginess of it out of it at earlier levels.
I allow them, but as a DM I also always use point buy or standard array, which means players often, but not always, opt for ASIs over feats.
I also don't tell new players about them. If players are engaged enough to actually read a PHB and want to use them, I'm happy to allow them. If it's something I as DM introduce without the player asking for it, it's a good bet that the feat will end up being one more ability that player will forget their character has and/or need me to explain over and over again during sessions.