Those all pretty much sound like kinds of spells to me...? An OD&D magic-user could do all of that. Reasonably confident an OD&D fighting-man with the right magic items could, even.
Dammit, KM, why can't you just acknowledge that my opinions are law and have done with it?
I don't see a shifter as being a spellcaster, necessarily. I see him as having one power that lets him change his form into things that have situationally useful abilities. I suppose you could call it a spell, but it doesn't really necessitate a spell /list/. In D&D, the druid has traditionally had a spell list, but that spell list has also traditionally been the cleric's spell list, so it's not really a point of differentiation.
Also, the wizard's spell list has traditionally permitted shapeshifting, but only in a limited capacity. The concept of a "transmuter" from the earlier editions of D&D depends less on shapeshifting and more on changing the attributes of other creatures and objects.
Regarding the mentalist, I'm not the biggest proponent of psionics in the world, but if they were to have a presence in D&D5 I'd like to see a return to a ruleset like the one from 1st Edition, where psionics had their own set of mechanics rather than relying on a variation of the wizard's shtick.
Finally, the summoner is characterized by having a permanent pet, which embodies most of the class' capabilities. Again, not necessarily a list-based caster. The summoner really has no current D&D equivalent, unless you count the ranger and druid from earlier editions, neither of which are really defined by their pets, or the summoner from Pathfinder.
If I want to be a healer/buffer without worshiping a god? Or to be a cleric without being a healer/buffer? Why isn't that OK?
To answer the question directly, it's absolutely okay; classes need flavor to fill an expository role, but that doesn't mean the provided flavor should be anything but a default setting. If you want to roll up a healer with no god, roll a cleric and give him a different backstory. If you want to roll up a caster who gets his spells from a god, roll a wizard and give /him/ a different backstory.
All I'm trying to say is that if you're writing RPG rules, and your system is class based, then maybe your classes ought to have rules impact. You seem to want a system where classes are non-binding, and I just don't understand how such a system is class based to begin with.