No, he was beaten out in camp by Gary Hogeboom one year. Landry went back to White later in the season.
You didn't give much detail in your original statement, so I assumed you were talking about the end of White's career, not the Hogeboom years. I don't see how this relates to your opinion that White only was the starter because he was on good teams and lost the job when he was on bad teams, suggesting that White was a coattail rider, which he most definitely was not. Injuries were always the issue with White. They ended his career. And they also led to him being benched (along with public perception that White could not win the big one) for a short time to Hogeboom, because had White not been injured, Hogeboom would never have gotten an opportunity to come off the bench in the first place.
I remember the quarterback controversy. Hogeboom filled in for White when he was knocked out of the NFC championship game against the Skins, and he did ok. Not great, but ok. But it was enough to start the rumblings for change. The fans were always tough on White at that time period due to the playoff failures, and because he was never going to be Roger. I was guilty of the same anger at White myself back then. The perception was that he couldn't get Dallas over the hump, and people were getting antsy as the team got older and older and older.
White played well the following season and the team went 12-4, but once again the Boys fizzled out at the end, getting blown out in the playoff game against the Rams. The calls for Hogeboom got louder, and that's when Landry finally decided to roll the dice and make him the starter the following season. I say roll the dice, because Hogeboom never beat out White on actual statistical, merit. He was a gamble and a hope to change things up for a fading team, but quickly it became apparent that Hogeboom was not the droids that everyone had been looking for, White was named the starter once more, and the failed Hogeboom experiment was over.
Danny White > Hogeboom
Danny White < Public perception and expectations
The end.