I do like the idea of the themed caster, I've done something kind of similar with an updated 5e Dragonlance mystic and sorcerer using the sorcerer class as the base. I've split the spells up into the various spheres and realms and, although a mystic/sorcerer can choose any spell from each of the spheres/realms (Mystics choose from spheres and sorcerers from realms), there are one or two abilities in the archetype that benefit them focusing on some spells from the same realm over a large amount of diverse spells.
Yeah, a themed caster, that's a good way of describing it. Wolfgang Baur's "path mage", the "domain mage" from Unearthed Arcana, and powers-oriented spellcasting systems (e.g True20) would be similar examples.
I felt that the question in [MENTION=6779993]Elderbrain[/MENTION]'s OP had a glaringly obvious answer (i.e. there are LOTS of ways to play a caster using healing and combat magic, from a bard to multi-classing to feat selection), so I assumed that the question was asking about something deeper. Or I assumed a deeper question because that's the question I wanted to answer
A themed caster does more than what could be possible as a bard, multi-classed character, or a wizard with a feat. And it also goes beyond the rather mechanical mandate of "a class that can use healing and combat magic."
I feel the concept of a class should NEVER be rooted in mechanics, rather it should be rooted in compelling themes, literature, myths, stories, movies, etc.
The basic design precept of a themed caster might be that "even archmages never truly master all magic, so you can be a master of one or two types of magic, a jack of all trades, or something in between."
But it would need to go beyond that to occupy a meaningful design space alongside the PHB classes. So it would need to provide features that were meaningfully distinct from other casters. For example you might include (a) features linked to each spell theme (e.g. the fire theme provides an ability to suppress flames around you), (b) a fatigue mechanic for how it casts spells instead of slots/day OR allow it to cast certain lower level spells repeatedly, (c) improved ritual magic casting, (d) something to do with True Names, etc.
Or maybe it would better fit in a 3-class system (Mage, Rogue, Warrior).