I’m sure a lot of people (like myself) would refuse to install a special custom TOTP app just to be able to use one site. They’re only hurting TOTP adoption by doing that, and thus increasing the amount of support resources they need to allocate to account hacking / hijacking.
I’m sure there are other companies too willing to go against their economic interests for various reasons, but they’re going to be in the minority.









I’m not sure what you mean by integrated into the OS. I use a FOSS password manager on android (keepassdx), storing passkeys both for apps and websites. When the browser or an app asks for a passkey, android uses the password filling service you’ve chosen (in my case keepass), and it loads up, no differently from when there’s a password prompt. I’m not sure how it works on other platforms like iOS, windows, or linux, but at least on android, you can use a FOSS password manager not integrated into the OS.
Specific sites aren’t locking down TOTP, this seems to be an EEE scheme of Authy specifically, since as you say the TOTP codes provided by sites still work with other authenticators. I’m referring to the case where say, amazon forces you to use their app to login, or creates their own custom TOTP or passkey app, and won’t let you log into their site without it, and then you have no control over your login method or codes.