These examples are what caused them to be this way. Hand waving was a lot more acceptable in mathematics while Weierstrass was alive. The discovery of clear counter examples to hand waving arguments lead to the desire to put mathematics on the strong footing it is on today.
It may annoy students today as there is seemingly little utility in these distinctions, but they actually are important. The more complex mathematics becomes, the more important it is to actually be on solid footing.
There was definitely some of that before. Ben Franklin, in his autobiography, writes that his Junto, a group of people devoted to self-improvement, included:
"Thomas Godfrey, a self-taught mathematician, great in his way, and afterward inventor of what is now called Hadley's Quadrant. But he knew little out of his way, and was not a pleasing companion; as, like most great mathematicians I have met with, he expected universal precision in everything said, or was forever denying or distinguishing upon trifles, to the disturbance of all conversation. He soon left us."
No doubt pedantic mathematicians had existed before, but what these examples did is actually convince mathematicians in general to be far more concerned about the subjects of mathematical foundations.
E g. an actually consistent and encompassing definition of what a "function" is, is something surprisingly recent. Certainly Euler, Gauß, Leibnitz or Newton did not have one. Only with Cantor's project of set theory did we get something remotely satisfying. Cantor faced significant backlash, which surely would have been much more effective if there weren't examples where actual foundational questions became questions of practical mathematics. And whether Weierstrass had actually found a "function" was one of these questions.
It may annoy students today as there is seemingly little utility in these distinctions, but they actually are important. The more complex mathematics becomes, the more important it is to actually be on solid footing.