I've been running Princes of the Apocalypse without feats or multiclassing for my university society. I imposed this restriction mainly to see what it would be like, and the players didn't argue in part because they were all new to 5e, plus I managed to sell it as an 'old school' element in the generally retro dungeoneering vibe of PotA.
The initial characters were as follows:
- Beastmaster Ranger (never turned up to play after char gen, guess he read online about the class?

)
- Beserker Barbarian (the party powerhouse, consistently does more damage and soaks more damage than any other character. Has trained the players to believe that Barbarians are the best class in 5th edition)
- Assassin Rogue (got himself killed by being an idiot in Feathergale Spire)
- Great Old One Warlock (deliberately not optimised at first, but now focused on using Eldritch Blast and various AoE spells to great effect. Wins fights for the party)
- Elemental Monk (got splatted in Sacred Stone Monastery)
- Lore Bard (player didn't have interest, got himself killed in Sacred Stone Monastery)
Replacement characters, for new and existing players:
- Swashbuckler Rogue (from SCAG. Cool character, consistently does well)
- Aarakocra Wild Sorcerer (total loose cannon roleplaying wise, but did lots of damage. Ended up dying to a fireball after a wild surge during combat had left him as a plant pot with 10hp. That was in the Fire Temple.)
- Cleric of Life (As usual for this subclass, his healing abilities are so good that he gets to spend a lot of time not needing to heal. Like all clerics everywhere, has fixated on Sacred Flame and Guiding Bolt)
- Battlemaster Fighter (this character, the third for the Assassin/Wild Sorcerer's player, has not yet started, but I believe that he plans to go Battlemaster
edit with a greatsword)
So there has been a fair variety of classes, not least because I have become death, the destroyer of characters. The players have not complained about the lack of feats or multiclassing, although as newbies you could argue that they would be less likely to. The party did have more interest in squishy classes than frontline types, at least until the Cleric replaced the Monk, so this might bear out the suggestions up-thread that STR classes would be less favoured. They also decided not to bother taking any healing classes at all, which went about as well as you can expect.
I've enjoyed not having feats. I would absolutely agree with the sentiment that feats tend to make characters one-note; in a previous campaign, the Warlock's player used the Trip combination Fighter (Battlemaster, Polearm Master, Sentinel) and it was pretty boring for me as the DM. Every time he did the exact same move, it usually worked, and it made combats less static. The Barbarian, for sure, tends to do the same thing every time as well - but it feels less gimmicky, somehow, when that is a basic attack roll.
No multiclassing is a thing that I will probably use going forward; no Feats is something that I will seriously consider as a house rule in my future campaigns. The Feats contain the most balance issues in the game, and they also contain the most dull combinations in the game. Removing them would seem an easy way to return to a fairly balanced field. I know that it could lead to charges of boring classes, but it is my intuition that people who complain about 'simple classes' would never play those classes regardless of whether feats were an option.
My two pence; I hope it helps, @
Zardnaar.