There are options besides "give every class access to every spell" and carving out healing as a special domain of magical endeavour. Rolemaster is one example.
Treating healing as special but not, say, teleportation; allowing wizards to have direct damage and battlefield control and enchantment and a reasonable range of buffs, but not healing; those are D&D legacies. There's no inherent design reason to bundle things in just that way, and other FRPGs - and 4e D&D - show other ways of bundling the various sorts of ability.
I mean, you keep asserting that there's no inherent design reason, but I've pointed out that there is: preserving class identity and balance. Rolemaster goes in the other direction - it's the system I first switched to when I got tired of AD&D. My experience with Rolemaster is that the granularity of character design actually led to more homogeneity, not less, and one of the things that led me back to D&D was that I decided I preferred a system with more distinct class identities that, in my experience, leads to a greater variety of player experiences.
Saying it's only there as a legacy is dismissive, and you haven't proven your case. Do you have some documentation of designers asserting that "gee, we just keep healing spells away from wizards because that's how it's always been?" I don't think so. I think you are just dismissing the possibility that it is good design to not give wizard-type classes much access to healing spells. I happen to think that it is. Your flaw is my feature. Ergo, good design from my perspective, and not just some legacy that no one has thought through. In fact, as you point out, 4e tried it differently. That didn't work out very well.
This is a theme throughout this thread: asserting opinions as facts and dismissing alternatives as, basically, driven by ignorance. Is it possible that maybe the designers looked at the feedback about 4e and decided that some of those choices turned out to not be great design for D&D?
The knee-jerk reaction, especially when someone doesn't like something as much as we do, is to assert that they just don't get it. They don't understand, or they're too cautious, or they are willfully ignoring the better option. Maybe they're just sheep. Or "mushrooms," to quote another poster in this thread.
It's super condescending. Maybe people are smart, and have all the same information, and have good intentions, and just don't come to the same conclusion. It happens all the time. Game design is an art, not a science, and what makes a good game is subjective. I think 5e is a MUCH better version of D&D than 4e. I think it is the best designed version of D&D. You have all the same information, seem very smart, and I have no reason to assume you have ill intentions, so it just seems like we are both reasonable people who disagree. That's okay.
But please stop dismissing my opinions as if I haven't thought them through and don't understand what "legacy" means.