But there are times when you want the entire party to stealth past a bored sentry - and being the worst in the party is a problem.
Then often the Fighter might be towards the bottom, but there is a good chance the Cleric is as well as any Paladins.
If the entire group needs to stealth past a location, you'll probably fail because someone will roll poorly. Spells like Pass Without Trace because almost a necessity if you want an entire party to stealth anywhere IME.
Frankly I feel it's not the number of classes - it's that wizards get access to everything (and clerics and druids do similar). My preferred approach would be to dump the wizard entirely - and add a number of sorcerer subclasses for some of the schools (especially one for necromancy). And if there isn't already one an illusionist bard.
I am not for dumping the Wizard class, itself, but I agree that having specialist schools (with non-access opposition schools) is better and want to see one of two types of casters (not just wizards):
1. the generalist. you won't have powerful spells, but you'll have a wide variety you can cast fairly often.
2. the specialist. your focus is going far and fast, you have very few spells and can't cast them often, but they are powerful.
To your point, a generalist might have access to a lot of different types of spells, but their spell progression would be delayed and many would focus on utility IMO.
As where the generalist would progress RAW as far as spell level is concerned, but would have maybe half the slots. Also, they would have restrictions on what they can access, like maybe just one to three schools.
Anyway, I am meeting to play tomorrow so I'll post about some of my discussion with the table probably Sunday.