There shouldn't be any guitar that's properly built that can't be made to stay in tune. I've had a number of Gibsons pass through my hands (some mine, some other peoples) and all of them have been OK with regards to tuning and stability once set up. Gibson did change their standard neck angle fairly recently, increasing it slightly, which meant that the pickups, bridge and tailpiece all stood a lot higher from the body. My mate's LP Studio with P90s had a ridiculously high bridge as a result, and I had to lower the stop-piece tail and put the strings in backwards and then wrap the strings over the top in order to get a decent break angle without the tailpiece being too high. I also wrapped PTFE plumbers tape around the insert threads of the bridge and tailpiece so that there was no mechanical play in the threads. Without that they did rather wobble about, which wouldn't help tuning stability.
On my 1995 Jimmy Page model LP, the tailpiece is screwed right down to the wood and the bridge isn't very far off the body. The tuning on that is as solid as a rock.
My 2014 Les Paul Melody Maker has a wrap-round bridge and that stays in tune.
Even my Flying V with a maestro trem stays in tune (after some fettling and using the PTFE tape).
Like Pablo, I also had a LP DC faded, which I think was a 2004 model which I bought used, early in 2006.
It's only real drawback was the thin whippy neck, rather like an SGs. Like an SG you could hold the body, shake the guitar and hear the tuning change. But it would generally stay in tune and sounded great. I gave it to my nephew for his 18th birthday.
Can you take a sideways-on picture of the bridge and tailpiece, Doc? As Andrew says, the break angle is important and too shallow or too steep (the strings shouldn't touch the rear edge of the bridge) can have an impact on stability.
Also, if you really loosen or remove all the strings, do the bridge and tail posts screws have a lot of slack in them? If so, it's worth trying my PTFE/Teflon tape trick. It's cheap to do and easily reversible. Doesn't seem to affect the string earthing either.