I have two guesses for why they need admin rights:
1. DRM code needs admin privs (at least that was the case some years ago). Possibly anti-cheat code also requires deep access to detect various nefarious software trying to help the gamer cheat.
2. Updates to the VC++ runtime modules and DirectX updates might require admin privs (but I'm fairly certain you can bundle VCRT stuff together with your executable in a place accessible to the user)
3. There is also the issue of sharing executables. Games are HUGE these days (50GB is "nothing" and it is considered a good thing to put these monsters on SSD units which aren't all that big), plus other users on the same computer are licensed to share games currently installed in your library.
The third option doesn't necessarily require admin privs, but the effect is the same: Users on the same computer are sharing executables and there is a whole lot of trust involved. (trust which doesn't extend to trusting people not to cheat and similar, but...) They might as well share spit and not wash their hands after using the loo.
My conclusion is that the end users want to run every executable he/she can dig up (anonymous usb-key found in the parking lot? Go for it!), whereas the typical admin type wants the user to pack up her/his things and go home (and never return). I believe it was the great BOFH who remarked that the perfectly running network requires the absence of all users.
Very disappointing response from Valve however.