Do you honestly believe the following; That given the stated responsibility, the House not only doesn't or shouldn't have the authority to investigate the Executive and impeach/indict it. And if you believe it does, how do they do that effectively if we allow the Executive to simply say "no" and deprive it the necessary information to do that smartly?
No POTUS in US history has so blatantly ignored the laws and customs. It's a dangerous precedent.
Yes, they have that authority, but that authority is not unlimited. "Executive privilege" is not something that Trump or Nixon made up. It's a critical aspect of the system of checks and balances. If Congress ignores it, and says that any time it is invoked they can and should remove the president from office, then there's no check on congressional power.
In reality, though, there is a check on congressional, and executive, power. If Congress makes a demand, and the president refuses that demand, then they can go to court to try to enforce the demand.
And plenty of presidents have asserted the ability to refuse to comply with congressional subpoenas. One of them, Nixon, went to court, and lost.
When the executive refuses a congressional command, the Congress could impeach him. That's what happened in December. However, they need 2/3 vote to convict. They'll never get it, and the public will yawn.
However, in 1974, the Supreme Court sided with Congress, and faced with that court defeat, public pressure mounted on Nixon. He and his lawyers decided that failure to comply would result in impeachment, so they complied. It turns out the material that was subpoenad contained enough bad stuff that public pressure still mounted, and the congressional Republicans turned on Nixon, and he was thrown out, or would have been if he hadn't see the handwriting on the wall and resigned.
The Democrats should go through that channel now. Refusing a Supreme Court order would be seen as a pretty big deal by the public, and it could turn at least some of the Republicans against him. Anything less is a major yawner.