Where did you get the idea that Apple is selling their AirTags for animals?
The AirTags are not designed for tracking a moving object ... whether its for pets or people. The were primarily designed to help you find lost or misplaced stationary items, like keys.
As you now know the AirTags rely on two things: 1) Bluetooth, and 2) The Apple Find My Network (which literally consists of millions of other iPhones.) Yes, your iPhone can only "see" the tag when it is within Bluetooth range (~30-40'). However, beyond that range, any nearby iPhone can relay the tag's ID & the phone's GPS location to the Apple servers over the Internet. Unfortunately, if you live in an area where there are few other iPhones, then yes, you will be limited to that Bluetooth range.
For pets there are currently two common means to track them should they get lost: 1) A GPS tracker, or 2) Having them "chipped."