Would Ezreal's magic be stronger if he actually stopped exploring for a while, and went through some serious training? I do understand that he would be able to cast spells without the amulet. But would he improve his performance with the amulet?

The amulet is a weapon, and it has its limits. Of course, he would be able to overcharge it and push it to the maximum of its power, but there are boundaries to what it can do. For comparison, no matter how great Vi's skill at brawling is, she can't go past the limits of what her gloves can do, so she has to constantly upgrade them. If Ezreal will truly shift to studying magic, which he has a great aptitide for, there would be a point where his own skill would exceed the amulet's capabilities.

He uses his magic somewhat like other champions use physical weapons, making quick moves and strikes with it rather than more intricate spells.

He could probably improve with less reliance on the amulet, but his speedy combat style seems well-suited to quick, easy to perform spells.

To my understanding, the purpose of studying magic is to learn how to manipulate it, control it, and gain access to the latent magic in the world. How to make sure that it obeys your command. Ezreal's gauntlet allows him to skip that process and go directly to his innate ability to direct the tapped magic. So I see Ezreal's gauntlet as a middle man, in essence doing the studying for him.

So, no I don't think his abilities with the gauntlet would improve if he were to quit exploring and start studying; however, I believe he would be able to use magic without the gauntlet if he were to do so.

THANIAS OUT!!!

Would going to Viktor and upgrading it help? Hextech Core Amulet OP

I don't think so. The lore seemed to imply that the glove gave him master over his magic without the need for practice, kind of like instinct.

We have no way of knowing all of his magical skills since he doesn't seem to employ them much at all. Lore-wise, other casters are described to be very skilled magically and that isn't reflected in their in-game skills.