I think you would technically be since what you agreed to by accepting the EULA is that you would have the game on your EA account and would rely on their services to play it, you don't legally have the right to play the game if it's fine into your possession another way.

And they fit sure have provisions about downtime and access issues in the EULA.

The digital area is something I haven't looked much into so I can't really comment on that but I know regarding physical media the relevant US laws only really make exceptions for things you've done yourself. Just because you own a physical copy of Pokemon Yellow doesn't mean you're allowed to download a copy of it from off the Internet. You're allowed to make and use a backup from a physical cart you own. This is why emulators can't (legally) include ROMs, ISOs, BIOS files, encryption keys, etc. as those are the copyrighted materials that you'll need to make a copy of yourself to legally use emulators.

To my knowledge (not a lawyer and this is not legal advice) what you did is indeed piracy because you downloaded it. If you had cracked it yourself you probably would have broken some licenses and whatnot that you had agreed to with EA, but I don't believe that would have been piracy.

Either way EA is very much unlikely to do much anything about it as for the most part the industry only cares about the sources of pirated materials. They generally only ever go after people distributing pirated materials so they'll (legally) attack torrent sites, ROM sites, and other such distributers. The most you're likely to ever get personally is a strongly worded letter (possibly a C&D) to your ISP from some AAA video game company if they notice you seeding a torrent for their game as then you're being a distributer of pirated materials.

Outside of that I've never heard of them coming after anyone for having the entire collection of GBA titles on their thumb drive or emulating Halo having never owned an Xbox or playing the latest Sim City without always online functionality. I'm not saying it can't or won't happen, but you'd make headlines if it did.