That's not really a Harry Potter influence though. I think there are two aspects of increasing prevalence of magic compared to older editions:For me, it's primarily the prevalence of magic in 5e. When nearly every single character is a caster, it's hard not to see the influence. Is it the only influence? Nope. Of course not. The specific source isn't really what I'm looking at anyway. You're fixated on the specifics of the books to a degree that I'm not. The fact that the game isn't set specifically in schools doesn'T really matter to me.
- More spells per caster. Old-school low-level casters mostly relied on mundane weaponry (primarily thrown daggers in AD&D, or light crossbows in 3e), but now they have the ability to cast equivalent-power spells instead. This started in late 3e with "Reserve feats", and continued with at-will powers in 4e and cantrips in 5e. I don't know if there's a specific source for this other than the desire to have wizards actually wiz. If anything, it probably comes from WoW, though at the time I believe the WoW at-will equivalent was the Wand weapon.
- More characters casting spells. This is more a matter of making everything beyond the ordinary into a spell. Rangers don't inherently know how to avoid leaving tracks when moving in nature, but they have the pass without trace spell. Totem barbarians can see through the eyes of beasts and speak with them, and this is represented as the beast sense and speak with animals spells. In 3e and earlier, these would likely be bespoke special abilities, or at most they'd be "spell-like abilities", but with 5e they instead went "Eh, we have a spell that does that, so let's use that."