Im actually able to replicate a multi steam.exe with no game launched, see below:
ToS requires a separate steam account for each game account (there's no login within the game, just uses steam directly), and the game also requires exclusive write access for each instance (hence the treeofsavior-1, treeofsavior-2, etc. setup). The problem occurs when actually launching the game itself (within steam, by clicking the play button), as the previously opened steam.exe clones will freeze and crash, causing the entire setup to malfunction. Here's what usually happens:
> open account1's steam.exe via innerspace (lets call this steam1)
> open account1's tos.exe via steam1
> steam1's tos.exe runs successfully
> open account2's steam.exe via innerspace (lets call this steam2)
> open account2's tos.exe via steam2
> steam2's tos.exe runs successfully
> open account3's steam.exe via innerspace (lets call this steam3)
> open account3's tos.exe via steam3
> steam1's steam.exe crashes
> steam3's tos.exe will show the "Steam is needed to run this game. SteamAppInit Failed" error
About the crash dmp file, for some reason I'm unable to find any after the fresh install. I'll do additional testing later to produce a better screenshot of the error as well as generate a dmp file.