Another option is to just store the keys on your ring and use them to interact with your mobile phone so your mobile phone never stores your private key

Sorry, but storing a complete private key on the ring is probably a bad idea. Since the merchant taking payments are surely funneling all payments via some bank or payment processor, they could just send your public ID and let the remote payment processor handle things. They could do an instant debit or even a delayed credit transaction. You could have traditional (and costly) option to dispute the transaction.

With banks testing contactless payments (NFC) for sums under 25EUR, the same could be done with NFC rings, but that would surely mean the ring must use other NFC chips and the whole process would probably get more expensive. The brave could perhaps dare to extract the active part of a Paypass and try to squeeze that into a ring. But it'd surely be illegal for the stores to receive a payment unless you have an unharmed card with your signature etc etc. So, trouble.