EULAs are a load of crap. We all know this. No one reads them, they can rarely be enforced and are worth nothing in law.

But they do highlight one thing: That you are entering into an agreement.

This is an asymmetrical agreement. Whether it's EA, Valve, or however else you get your games, you are getting service from the service provider. And they are king. (Of their service that they provide).

You have no rights to recieve this service; its provision to you can be terminated at will, without reason.

And that happens. You've heard of the EA bans and the Steam bans. These companies do not have to be reasonable. They do not have to hear you out.

Moreover: These decisions are not made by large, (in)fallible organisations. They are made by people. Customer service people sitting in front of a computer that can ban you at will. No matter if they misunderstand things or get things wrong; they're just doing their job.

All that stands between you and the loss of all those games is some guy just doing their job. You have as much chance of being a 'false positive' as everyone else. You cannot assume that these people are gonna be reasonable with you.

The take-home is that:
It doesn't matter that the EULA contains astoundingly arbitrary reasons for possible bans. Because if they ban you, it can be for anything they like, EULA or no EULA. As arbitrary as they like (and, moreover, they probably won't discuss why they banned you with you)

You can do two things:
1) Hope that you won't ever fall foul of some guy doing their job.
2) Backup your games

Get a large, cheap, external HDD. Whenever you download a Steam/Origin/Greenman/Gamersgate/Gog etc. game, make a backup. Then, whenever Steam, EA, or a game-download site ban you (or go bust) you can just crack your games and play them.