The problem with this is that the King has absolute rule. He can, theoretically, kick you out or carve up your lands at a whim. Obviously, there is a vested interest for a King to NOT do this as he often needs support from those same Houses, but in the prime Targaryen era with dragons, they kinda can do WTF they want.
So trying to play neutral will just get you attacked by one side or the other and cut off any support from the Crown. Kings often need lands and titles to dole out to supporters so if they see a House playing both sides that's the place to threaten in order to get their direct support. Barring a vast geographic distance (like the Starks and the North in general) most Houses can't sit things out for long lest they get hit themselves.
Kinda disagree.
It's not playing neutral to just keep to your own lands and make sure they're running well, with disputes quickly settled and adjourned.
One of the reasons why Tywin fell out with Aerys was his constant push for Lannister advancements and the repute he gained with his administrative skills.
Which became entwined and eventually made Aerys realize just helping to run the realm wasn't enough for old Tyty.
And Oldtown at least is very remote in comparison to the Red Keep.
As long as the Hightower stands and burns bright, what can they really gain from being in the mix?
Older than everyone else and richer too?
All you're doing is putting a target on your back with every lesser lord/lady begrudging you your history and prestige.
We can agree to disagree but my stance is its better to maintain and consolidate your ancestral holdings vs get greedy by being in the mix.