I think there are other options too
Notably they now support both the Apple and Android ecosystems from a single hardware SKU, although only one at a time, and the formerly disposable battery can now be recharged with a standard Qi wireless charger.
Having recently switched to Android I was tempted to give Chipolo products a try, but this reddit thread disuaded me, as multiple users there are reporting the exact kinds of issues that I experienced previously: https://www.reddit.com/r/Chipolo/comments/1n4m4j3/chipolo_po...
Any recommendations for a rechargable but thin one, AirTag itself is too thick for regular wallets.
I've been using one of these with Find My for over a year. I've not done anything with the Eufy app, just the Find My app itself. It lacks the precision find, but you can play a sound through the Find My app / get directions / share the item / flag the item as lost / do left behind and it works well.
I own 8 AirTags, and have them on all my sets of keys and in all my bags. I've managed to avoid loss about 5 times in the 3 years of using them. It also gives me piece of mind when landing on a plane that my luggage is where its meant to be.
If you want to stop your wallet being stolen, I'm afraid your options are very limited.
As for theft - you might be overthinking it. It's a black piece of plastic that could easily be a door key. Gutter thieves will rifle the wallet and pull out cash, and credit cards, and drop the rest in the trash anyway. Any regular thief knows they don't want to be caught holding someone else's id.
I think there are also several third-party wallet trackers that integrate with the Find My network.
Have one in my wallet. Will probably get some more AirTags.