It is not a DRM problem, you can run many EA games on Linux with no problems, it is an anti cheat problem, which can not be solved by Valve, it has to be done by EA.
Valve can not solve it. The only way it can be solved is if game studios create anti cheat software, which are effective and can be used within a Linux environment. This will only happen if companies see a profit motive to do this, which will happen if the market is large enough.