I'll just say...
(wait, I want to make sure I'm not near anything that might attract lighting)
...that Max has a point here. Or, at least, there is a point that can be extracted from Max's posts.
If you want a fantasy world in which wizards are rare and mysterious, you need some mechanism to explain why that would be true. If magic is as "easy" as 5e makes it appear to be, you would have a steadily increasing numbers of Wizards. After all, as
@Tony Vargas would happily tell you, Magic > All in 5e. And even if not, it's awfully useful and powerful.
You could, for example, simply rule (as a DM) that only some people have the magical spark. Players get to have that spark if they want it, so no random roll is required, but part of the fictional setting is that not everybody has it. But that's not part of the actual rules anywhere.
Or you might rule that Wizards are, effectively, a guild, and maintain strict control over their trade "secrets". This is quite reasonable and feasible, in my mind. Not so different from the way certain swords, produced not by sole individuals but by secretive "guilds" (in deed if not in name) were so highly prized over the centuries. Whether Uthbert, Damascus, Toledo, or whatever, the smiths who made these blades taught their apprentices how to do it, but as a group maintained an iron grip on the trade secrets. (As an aside, it blows my mind that they even figured this stuff out, and how to replicate it, with zero understanding of the actual metallurgy and chemistry.). Wizards might do the same thing.
Or maybe it's just that teaching magic sucks up a tremendous amount of time. The Wizards aren't "secretive" so much as just really uninterested in spending 10,000 hours training somebody.
Whatever it is, if your world does not have an explanation for why magic is rare, I think Max is right in that you're either going to have more and more and more wizards (and maybe bards and clerics and what-not) or you're going to just have to live with an incongruity in your game world, similar to how it makes no sense that you can't easily and cheaply buy copying privileges to fill up your spell book, since there's no real "cost" to letting somebody copy your spells.