That would leave the ranger as the only divine caster, getting rid of wizard, druid, paladin, and cleric, making the bard the main healer and the warlock the most priestly-type character. A whole lot of spells would exist as magical secrets for bard only, although the divine soul sorcerer technically covers all of the cleric spells.
that would be a very cool campaign premise, but a clear departure from the base assumptions of a typical D&D game.
Divine vs not divine spells is fluff and the priest background over-rides "priestly" classes for anyone who wants the fluff. Religion is also easy to tie in regardless of background.
I've played campaigns banning magical healing too. It's also unnecessary in 5e because of the healer feat, potions on the equipment list, and hit dice healing on short rests. Plus the various class self healing that may be available.
If a group does believe they need a dedicated magic healer then bards work and divine sorcerers as an alternative.
It plays just fine.