I am a little confused you say "the problem is that Steam isn't running," but you describe the issue like steam is running just not launching games.
Secondly try changing the setting in setting, like turning off esync or msync or DVXK or on if their off. Or changing Wine from crossover to GPTK and seeing if CS 2 runs. There might not be a setting that will work for every game, so you might need to change setting now and again. There was over with Whisky.app ( now defunked) a setting for CS2 that might still work now, not harm in trying. Hope this helps.