When one branch decides they don't like something, a second branch says they won't lose sleep over it (a recent quote from a congress-slime), does it matter that the third branch says "hey, you can't do that"?

Serious question. Not trying to be argumentative or hyperbolic.

Partially because the Executive branch needs the judicial branch in order to do a lot of things. They have to use the courts in order to get other entities (private parties, other governments, etc.) to comply with their rules. So let's say there's a dispute about whether a member of a government board has or has not been fired by Trump: he gave Oldie Oldguy the boot, and replaced him with Novice Newguy. Court rules that the firing of Oldguy was unlawful, and orders that he be reinstated - but the Administration refuses. The court doesn't have troops, so they can't physically force Newguy out of his chair on the dais or to have Oldguy's access card turned back on. But when the Administration tries to enforce any order, decision, regulation, or resolution passed by the Board against any other private party, the court hearing that case can just find against the government on the grounds that the Board wasn't lawfully constituted.

And partially because even if the Executive thinks they can get away with flouting a court order, no one else will act that way. So if a court orders the Administration to pay $X to party Y, and the Administration chooses not to comply, the court has the ability to order the seizure of federal assets to give to party Y. The government's money is stored in various places, but generally it's in the Federal Reserve Banks - which are quasi=private entities that aren't directly under the control of the Administration, and they are going to obey a court order to pay funds.

And finally, it's a massive political undertaking to defy a direct order of the Supreme Court (which is where a lot of this would end up) - and that might be a bridge too far for any Administration to really survive and be able to function politically. Trump has all the attributes of a bully - he relishes fights against weaker folks he knows he can dominate that don't have strong defenders (like USAID), but he generally doesn't like fights that have the chance to really hurt him and will quickly back down rather than take a major risk (like staring down a market downturn to get his tariffs). It's fun to make fun of Congress as not having a spine, but the truth is that Trump has been very smart in his early targets for DOGE - he's gone after agencies and programming and spending that is very unpopular among Republicans. But as soon as he started moving into agencies and programs that the GOP cares about, like Veteran's Affairs, Congress started sending him and Musk some not-so-subtle signals that he is venturing into a bigger problem area and that they want some answers on what's going to happen.