Is there really a market for Teddy Bridgewater?

He did a good job filling in for Brees. That's for sure.

But my point was to list him as your #1 starter is different than bringing him in as a backup.
Minnesota keeping him and not having him as their #1 would have been a demotion for Teddy. So either Minnesota had to sign him as their #1 or not. They chose not to because they wanted to try to find a better option. Whether that option is better or not is now hindsight and still is at best debatable.

The Jets signed him and moved on quickly and didn't keep him as a "mentor" for their new QB.

I think Teddy is building the case that he could be a starter again, but any team that goes with him as their starter will likely have a strong contingency plan (or I feel they should). That's just my opinion.
The Jets are the Jets. What's your issue with him outside of the injury and why would you need a contingency plan?