They won't make it illegal, there's no point. But, it might become impractical.

Imagine this, you have your whole desktop online, you pay for space as you use it, you don't need to buy drives, your space just expands for your needs. Do you want to play games just for a short time? Well, you can just pay for the time when using a vGPU. In fact, a game could even make requests for a certain amount of GPU power, and you get billed accordingly.

Everything is on the cloud. So, what happens then?

Well, people would instead of buying a PC just buy some low power ARM based terminal. In fact, it's likely many cloud services would include one at no extra price in the contract.

When the majority of people are using this system. What happens to the now "niche" home computer builders? Well, first parts will become expensive. Making less of the home based stuff will make it more pricey. Choice will become limited. It will be prohibitive to build your own. But also consider that online games might well also prevent you playing on your own hardware. It's far easier to cheat on your own hardware than it is on strictly controlled cloud hardware.

I don't think it'll come to this, or if it does, it won't be for quite a time even though this is entirely possible now (look at shadow PC for example). But if cloud computing in this form did take off in a big way, this is how it'd likely be and there'd be no reason at all to prohibit home setups. They would just become less practical and affordable all the time.