(Inspiron 17R 5720) LCD bezel - rubber breaking down and getting sticky

It's a moot question -- you can't replace just the seal, so you're in for a replacement display assembly if you want to correct the issue.

Whether it's worth replacing the display assembly on a 10+ year old system is a question only you can answer.  Carefully compare the cost of complete used systems -- perhaps a few years newer than this one -- before committing to the repair of this one.

Are you sure it's the rubber and not any liquid coming out of the LCD screen?

No liquid. Just the traditional rubber sticky and breaking down.