I would attach contact paper to the back of the mat to hold the backing together.

All depends on how much the mat cost originally. If it was expensive it might be worth looking into repairing it. But it might cost more to fix it then the to replace it

Depends on the value & quality of the playmat how much energy and costs I would stick in it ! Estimate the energy/hassle and cost you have to spend to repair it and google the price of a new one, then make your decision.

I would maybe look into sticking a new rubber backing on it. If test didn’t work, I would toss it as I would get sick of cleaning up the crumbled bits of rubber

Agree with the caulking gun!
Have a look on YouTube, there’s a tutorial aswell

There are a few ideas on Pinterest, but if you have a caulking gun with the acrylic tube, pipe a few lines of that. Your mat won’t go anywhere!