Why do I need to use the oven to make crème brûlée? Or do I?

There is no magic about the oven here. It is really for convenience. There is no reason your method will not work. I've made custards using a water bath and an immersion circulator (sous vide) many times...no stove or oven.

You can certainly use the stove, it is just that using the oven works better.

• on the stove, you will have to babysit the water bath so it doesn't get too hot. In the oven, it is less likely that this happens, because the oven heats up slower, and if it starts getting too hot, you will have more time to notice and react.
• the oven will heat your custard slower, resulting in better custard.
• the oven heats from the top too, resulting in more even temperature throughout the custard.
• if you are cooking with other people, somebody could accidentally stir your custard on the stove, maybe mistaking it for a creme patissiere or another type of stirred custard.

And in general, even if it there were no technical reasons, it is just normal to use the oven for the "stick it in and forget it for several hours" dishes and the stovetop for the dishes which need a concentrated blast of heat. So it would be natural to see this kind of recipe to be written for the oven, instead of trying to find workarounds to make the stove do the job which the oven was invented for.

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To address your update, you didn't make cheese, your creme brulee curdled on the bottom. This means that you didn't manage to make a setup that heats low enough (point one) or evenly enough (point three). You will have to fuss more until it starts going right.