The string of poor drafts really began in 1980 rather than in 1976. In '76, Dallas drafted Butch Johnson and Tom Rafferty, a very good WR and OL. They also drafted Beasley Reece who became a good DB for the Giants. Brandt didn't cut him, presumably. In '77, they drafted Tony Dorsett, Tony Hill (the WR), Andy Frederick and Jim Cooper, two good OL, and Steve DeBerg who had a long career as an NFL starter/backup. In '78, they took Todd Christiensen who balked at being moved to TE. He went to the Raiders and became an all-pro TE. They also took Dennis Thurman.
In '79, Robert Shaw, the first pick, would have been a great center but for a career-ending injury. And they took Doug Cosbie and Ron Springs. They also took Garry Cobb who became a good NFL LB. Again, Brandt didn't get rid of him, presumably.
Trouble started in 1980, and some of those drafts are hard to defend. Really, the drafts were poor through 1986 (the best picks being Jim Jeffcoat in '83, Eugene Lockhart in '84 and Herschel Walker in '85). Those drafts led to the disaster that brought down Landry. There was a little bad luck (Billy Cannon was going to be really good, as was Mike Sherrard) from injury, but all teams have that. There were signs that Brandt and company had adjusted. The '87 draft brought in Kelvin Martin and Kevin Gogan. The '88 draft yielded Michael Irvin, Ken Norton and Chad Hennings -- even David Widell, a good NFL OL the Cowboys let go. But the era had passed.
Years ago, either Brandt or Schramm said the biggest mistake the Cowboys made was in continuing to take chances after the rest of the NFL had caught up to their sophisticated draft model. They continued to try to force boom/bust picks, but while the other teams were behind, many of those picks were boom. As other teams caught up, most of those picks were bust. I think that's probably true.
Bottom line -- Brandt was instrumental in creating one of the great runs in all of sports. From the mid-'60s through the early-'80s, the Cowboys were almost always good to great. Brandt should have been in the Ring of Honor a long time ago. He really did change the face of the NFL, and he picked some of the greatest players who ever lived.