Repairing old luggage

I just bought this super cool, vintage travel bar for a future epic road trip. Unfortunately, the elastic that holds everything in place is old and stretched out, or even ripped. This will not do for my gin-swilling friend and me.

If I take the black shell out of the metal case, I don’t think that it will go back in the same way. Does anybody have an idea for replacing the elastic without access to the backside of the attachment points?

If you can manage to carefully cut or drill out the rivets, you should be able to replace the elastic and fasten it with blind rivets of appropriate size.
Somewhat uglier method: cut the elastic, but leave an inch or so on both sides of the attachment points; then sew new elastic onto the stumps.

That is perfect! I grabbed my Dremel tool and can grind/drill out the old rivets, and found a co-worker with a riveting tool. Thanks, Idnotus!

Since your friend is familiar with pop rivets (or any kind, really) you should ask him about “backup washers” or using small fender washers to spread the load over a larger area of the elastic straps. This will help prevent the rivets from pulling through the new elastic strips.