Being old, I have some sympathy for the OP's position, though it cries for nuance. Recall the book "Next Year's Champions," which was published in 1969. The title of the book wasn't meant to compliment Tom Landry or the Cowboys. The Cowboys had been the heir apparent to the Packers, having lost back-to-back NFL championships in agonizing fashion. The Packers were dead and at the front end of a lost decade. The Cowboys were heavy favorites to be in Super Bowl III. Dallas was loaded. They finished the 1968 regular season 12-2 and in a 14-game regular season outscored their opponents by 245 points -- 17.5 points per game. Then they laid an egg against the Browns. In 1969, the Cowboys finished the regular season 11-2-1 after having added Calvin Hill, dubbed "Super-Rook." Once again, they laid an egg against the Browns.
Whether from a broken heart or a broken body, Don Meredith had retired after the 1968 season, many fans having turned on him. Craig Morton had seemed an ideal replacement and was something of a golden boy. The Dallas defensive line was Lilly, Larry, Jethro and George. Jordan, Howley and Edwards were the linebackers. Renfro and Green were in the secondary. The OL was loaded and included Rayfield, Niland, Nye and Neely. In addition to Hill, the backfield featured Garrison and Reeves. Hayes and Rentzel were the WRs, Ditka and Norman the TEs.
Landry was under enormous pressure -- can't win the big one. And he was slow to install Staubach, though the Navy grad was spectacular every time he stepped on the field.
In 1970, the Cowboys started the season 3-2, intensifying the pressure, and Landry still couldn't bring himself to commit to Staubach who he considered a bit of a wild card and didn't trust to adhere to the game plan. The Cowboys went to the Super Bowl for the first time, but lost perhaps the sloppiest of Super Bowl games to a Colts team that was perceived to be old and on the way down.
In 1971, Landry still hesitated to start Staubach and actually alternated him and Morton, including in the infamous shuttle system versus the Bears -- the quarterbacks alternating plays. Dallas staggered to a 4-3 start to the regular season, and Landry made the decision to move to Staubach as the full-time starter. In three December regular season games, the Cowboys out-scored their opponents by a staggering, combined 89 points. They closed the season with 10 consecutive wins, including three in the post-season, which culminated in the Super Bowl win over Miami. Next Year's Champions became America's Team.
The point isn't to second-guess Landry re his hesitation with regard to Staubach. Nor is it to suggest that Garrett will enjoy Landry's success. Tom Landry was one of the greatest coaches in NFL history. The point is there was much more pressure on Landry in those days than people who didn't live through it are likely to understand. Yes he had pulled an expansion team from nowhere to the league's elite. But the post-season losses -- especially those two shattering losses to the Browns -- took a great toll on fan support of Landry, and fans in general wanted the move to Staubach to happen much more quickly than it did.
If there had been an Internet forum, I find it difficult to imagine there wouldn't have been a great groundswell of angst and insistence that Landry be replaced. Having lived most of the Cowboys' history, I would suggest those two Browns games remain the most shocking defeats the franchise has suffered. Younger people can't imagine how stunning it was to see Dallas lose those games. The Cowboys were perceived to be loaded, and they laid consecutive eggs. And the Colts Super Bowl -- suddenly it seemed Dallas might have missed its window. Some of its stars were getting older. Then the Duane Thomas problem blew up... the Ezekiel Elliott situation times 10, let's say.
So I would never presume to say individuals who populate this board would have called for Landry to be fired. I have no way of knowing that. But there was at that point a great deal of fan and media angst. A lesser man than Murchison might have fired Landry.