I'm not sure that's on the House though. There were no specific and narrow claims of immunity or privilege to litigate here. The WH argued that the lowest level employees were off limits completely. If you pierced that broad shield, then you'd have to deal with the narrow and specific claims that would follow.
Regardless, imagine that SCOTUS fully rules in favor of the WH here - essentially allowing the WH to claim total immunity from congressional subpoenas. The House could STILL decide that wielding that power as done in this case is worthy of impeachment. In other words, the House decides the pain threshold regardless of where SCOTUS decides legality.
Remember, the courts don't like to settle these kinds of disputes. It's not an irrational position for the court to ultimately determine that the power of POTUS to Obstruct Congress is precisely the leash that Congress decides to give the Executive. This is actually a rational position to take in order to balance powers.
In other words, a proper balancing might be for the WH to provide the bare minimum level of cooperation it needs to avoid the obstruction article. The Court is not going to decide where that line is, even it makes technical rulings on what privileges apply to whom and when. From that perspective, Congress is setting precedent here that an administration wide claim of total immunity is Obstruction of Congress and impeachable, regardless of how courts might rule on narrow and specific claims of immunity/privilage.
I think that's a rational position for congress to take in effort to maintain it's powers. The rational position for the WH would have been to cooperate at the minimum level necessary to avoid the Obstruction article (unless you think your guilty on the primary charge, in which case you might decide obstructing is more rational).
I get your point, but at the same time the way this is working basically shows that while obstruction is an impeachable offense, there is no standard on which to measure it and there are no consequences to doing so. Trump was impeached, well whipty frigging doo. What did it matter? Congress has already virtually neutered itself over the last several decades and now when they decide to make a stand, it didnt amount to anything. The president may as well be a king, because they ceded their power of how to spend money, whether we attack other countries, and now they've undermined their ability to hold him in check. What is there to keep the next president or any of their successors from doing whatever the heck they want?
We have a supreme leader with a bunch of monkeys at his feet fighting over who gets to be the head monkey.