Mid-length gas on a 12.5" would be equivalent to rifle-length gas on a 16" gun. Could it be made to run? Sure. Would it take some work? Probably. Is there any benefit? Probably not. Would it be worth the trouble? No, not to me it wouldn't.
To the original question, were I going to own just one AR, and was willing to deal with NFA for that one AR, and was willing to accept the concessions from doing all of that, I'd have a 12.5" Noveske stainless with Switchblock. It wouldn't do any one thing perfectly, but it would do a lot of things well enough. I'd put a 1-6x optic on it and call it good.
To that end, for a "patrol rifle", that might be a good option as well. If a can just isn't in the cards for a patrol rifle, skip the Switchblock. If you're not going to put a magnified optic on it, skip the stainless and get the N4 profile hammer-forged barrel. If you're only going to run it suppressed, get the 10.5" barrel, no Switchblock.
Expanding the scope a bit, for folks that have a 16" gun already, I think the 11.5" barrel is the best compromise when running without a can. Short enough that you get some benefit in the weight/length department that you can actually appreciate, and long enough to that you're more likely to get better reliability. The 12.5" is too close to 16" for my tastes, and the 10.x is still (even in 2015 when we can make 10.x guns run really well) a bit of a ragged-edge gun. 10.x,mhow ever, starts to make sense if you're doing a dedicated suppressed gun, and I'd do an adjustable gas block to get it tuned initially and then "set it and forget it".