True, but vast majority of websites don't do the right thing. So password managers need to manage a database of form input overrides. I worked at a small company building a password manager and it was a bit of a nightmare, we had to allocate support staff resources to handling reports of website incompatibilities.

I'm curious, why did you implement your own password manager?

I worked for a small company that did. The idea was the password manager lived only on your phone, and you'd connect it to your computer in various ways. Notably including our own hardware, USB nub that connected to the phone app via Bluetooth and acted as a keyboard for the device and the app would instruct it to do the keystrokes to enter your saved credentials. Also a browser extension that the app would connect to over websockets (through our relay servers, DH key exchange to encrypt the connection end to end) and the extension would request the app to send down the credentials on demand. Also we were a very early adopter/implementor of FIDO U2F (security keys). The product kinda hinged on the paranoia of the userbase (which I never aligned with, I trust 1Password and cloud sync'd encrypted vaults for example) so it never really took off. It wasn't "my" product, but I worked on it for a few years, right out of school.