MacBook Pro 2013 NVME booting Issues
You have pretty much answered your own question.


Tried SSD on another system and it works, CHECK
Old drive works fine, so it's not the Mac, CHECK


This points directly to the only other piece of hardware you haven't tested elsewhere, the adapter. So, unless you can test that adapter and drive in another similar Mac I would say you should replace the adapter.

Note that I have been doing computer support (Macs, PCs) for a long time and have learned that nothing is always as simple as that. Your issue seems cut and dried, and if I was a betting man I would bet on the adapter being the culprit. But computers are complex beasts and don't always yield to logic. They're too complicated, too many parts interacting in too many ways, for everything to be bug free.