I'm currently playing a Sorcerer-Warlock. Any spell he can cast, he can cast again after an hour's rest.
The difference between healing with a long rest, and casting healing spells five times with short-rest recharges, is that you can only Long Rest once per day.
There are many magic systems which don't have "max spells cast per day" as a limiting factor. Mage uses Paradox: the more often and the more blatantly you bend reality, there are increasing odds that reality will snap back, and give you a Time Out from casting further spells until it's calmed down and cooled off. (Possibly with other consequences too.) Shadowrun uses Drain: spellcasting can hurt, like exhaustion or wounds.
Harry Potter is fun to read but it's not a good model for "what would wizards do, if they had PC levels of ambition and creativity, plus zero-cost spell-casting." I mean, he gets a device which can roll back time three hours, and he only uses it to spend more time doing homework. He doesn't realize that any time he loses a game of Quiddich, he can turn back to the beginning of the game, and *start the game already knowing where the Snitch is*. Wizards have gold coins and silver coins, even though they can transmute silver into gold. Wizards didn't eradicate smallpox - they left that to Muggles; every death from smallpox from the 1870s to the 1970s, is a death Dumbledore could have averted. The list goes on and on.