I think you're underestimating how much that would cost and how long it would take. Every single game concept and execution would in fact need to be run passed the legal department. If a single steam employee just glanced at the game as said "god no, we won't approve a game called Active Shooter" then what sort of crappy "approvals" system would that be? It would be a complete crapshoot, since one reviewer might not have a problem with that, while another would. So you could just re-submit the game and cross your fingers that you get a different guy this time. If they're keeping any sort of per-game records to prevent people just doing that, then it would end up as mountains of paperwork, not a quick glance at each game.
Also the legal and PR implications of taking responsibility that way, and the PR implications of having an approved/disapproved system and how that would play out with social media battles over which games did and didn't get approved.
EDIT: "We played it for three minutes" isn't any sort of approvals process, and you could face legal liability then. Being able to say "we don't control what goes up" is in fact a good defense against being sued for a game containing something like a racist caricature at the 10-hours-in mark.
EDIT2: Imagine a hypothetical situation where the game flashes up child porn at 11 hours, but your budget and rules only specify that you test each game for 10 hours. Every dev would learn the 10-hour limit, and malicious devs could work out how to exploit the approvals process to slip things by the Steam censors. And some asshole would just for the lulz of what they managed to slip passed the Steam game approvers. Then, since it's "Steam approved" making the company legally liable becomes a more plausible court case.