The Trump Administration's argument against enforcement of the subpoenas is legal malarkey.
https://www.theatlantic.com/ideas/archive/2020/01/trumps-defense-against-subpoenas/605635/
To wit:
"Therefore, under the Constitution, Congress has both a responsibility and a right to inquire closely into the operations of the federal agencies, programs, and employees it authorizes, regulates, and funds. The power of inquiry includes the power to use subpoenas to compel production of testimony and documents. The Supreme Court has repeatedly recognized this 'oversight power' and held it to be 'coextensive with the power to legislate.'"
Also:
"The 'sole power of impeachment' granted to the House of Representatives by Article II, Section 3, would be meaningless if the House could not compel production of the evidence necessary to determine whether impeachable conduct had occurred."
The best precedent for this is the Supreme Court case U.S. v. Nixon (1974), where the Supreme Court unanimously ruled that Nixon had to turn over the Watergate tapes. Nixon resigned shortly thereafter.
https://en.wikipedia.org/wiki/United_States_v._Nixon
That the Trump Administration took these subpoenas to court is irrelevant. They had no leg to stand on. The tactic was to stall out the clock, and it worked. They basically just ignored them and shouted "Executive privilege!" into the microphones and on Twitter.
Obstruction of Congress is a thing and it looks exactly like this.