Why are you not gruntled? 
If Nixon had burned the tapes (as he later wistfully told a friend he wished he had), I think Congress would’ve assumed the worst, that there was something even worse on them than there really was. And what was there was plenty bad enough - Nixon conspiring to block the FBI’s investigation of the Watergate break-in, the “smoking gun.” If anything, the impeachment process might’ve shifted into high gear, esp. if Nixon had destroyed the tapes after Cox had issued a lawful subpoena for them.
The tapes were the evidence that required impeachment. I think the OP is making a cogent argument that Nixon’s best shot at avoiding impeachment was destroying the tapes. However, that was prohibited by Nixon’s personality.
Also, remember that Nixon had a lot of enemies in his own party and a lot of natural allies hated him: The Washington Post and The New York Times were Establishment newspapers and still are. But Nixon managed to alienate them and movement conservatives like Barry Goldwater.
You’re nuts. Destruction of evidence under subpoena is a felony. Publicly destroying the tapes, after their existence was made public and they were under subpoena, would have brought immediate impeachment and near-unanimous conviction.
Yes, Nixon challenged the subpoena on grounds of executive privilege, and some Republicans supported that challenge. Regardless of their personal feelings, however, all Republicans felt that the appropriate forum for resolving such a challenge was the Supreme Court.
If Nixon had quietly destroyed the tapes before their existence was made public, and before they were under subpoena, he might have survived. However, keep in mind that before the tapes were aired–that is, while the Supreme Court was still considering executive privilege–a large bipartisan majority of the House Judiciary Committee recommended Nixon’s impeachment, based on the testimony of underlings before a grand jury. He just possibly might have survived in the Senate, with only 34 votes needed for acquittal.
Again, the House Judiciary Committee recommended impeachment (by a majority as large as 28-10) before the tapes were heard.