Hi and welcome to the community,
I am using 2FAS. It has a browser extension, but the secrets are stored solely on the phone. The extension receives the generated code from the phone, requiring user’s confirmation on the phone. The developers have stated that they will never develop a desktop version due to security issues associated with desktops.
Yes, Authy has removed all its desktop applications.
If I were to select a new authenticator now, my personal preference would not be BWA, because the encrypted backup may be iffy, depending on your phone manufacturer. A Pixel phone would likely have the highest chance of conforming to Google’s security model.
On the other hand, if you are already using BWA, it is on the roadmap to sync TOTP secrets between the password manager and the authenticator. This may be a convenient feature for some users, but not to me.