I'm not sure how Garmin works, but for instance with Google Wallet-compatible watches, you need a phone where wallet can run. I've had this setup for a year where I loaded the cards from another phone and used a watch to pay.
However Wallet didn't like this setup. Tokens expired at varying delays, sometimes a day, sometimes a week or payment failed without reasons.
Nowadays, I just use my bank's app which work fine on GOS.
You only need a phone to add the card to the watch. After that it works without a phone.
I was actually very surprised Garmin supported the country I'm in. They don't even support the language script, I get squiggles, but payments - better than Google Wallet.