A typical 16-hour day...nowhere for the pilot to rest. He or she will be sitting near a gate, in uniform...trying to shut out the noise of thousands of passengers walking by and hundreds of public address announcements.
It seems like a quick-and-easy solution would be to have the Pilots' union demand access to the airline's network lounges (or better yet, make it an FAA requirement so that they can get into off-brand lounges at smaller airports) in their next collective agreement. Union members would commit to not taking advantage of the free food or the computer terminals, and the airlines would in exchange let them hang out there as it's quieter and gives them somewhere to get a bit restful before their next leg.
And if the airlines don't want to fill up their own lounges with free staff (as opposed to passengers), contract out the problem - give the pilots PriorityPass memberships, so they'd be going into lounges that aren't their own, and that are explicitly OK with people paying to get in.