Most operating systems and browsers come with a sync mechanism that many people default to, but it's no more than that: the default.
As for account recovery, most websites have a way to recover your account when you lose your password, there's no reason why that wouldn't work for passkeys. Every website with passkey access I've used so far makes passkeys optional and forces you to set a password already. If they switch their default to passkeys and add a password as an optional step, nothing would change.
Assuming the FIDO Consortium approves. Otherwise you might be like KeePass, being unceremoniously threatened with revocation of attestation, for offering to do what the big players are able to do.
When I'm on non-Apple I can scan a QR code on my phone and it will validate it the same way, using bluetooth to send enough info to the target computer to get me logged on.
It is not true. You can move passkeys between OSs if you have a password manager or an OS that has this ability. For example, I store my Passkeys in iCloud Keychain and I have them synced on all my Apple devices.
There are cases though, where the website can force the requirement of a device bound passkey, but that is something it is not very likely you will encounter.
And I suspect that most geeks, hereabouts, could set up their own signing system.
It's not all sunshine and roses, though. Despite having Bitwarden set as the only passkey provider in my Android setup, the phone persistently only offers me Google. Which is empty, because as I said, I won't use one tied to things I can't control. Works great on desktops, though.
Passkeys can theoretically require you to be on hardware, I haven't found anything yet that requires that.
They work fairly well. I use a login ID/PW to set up the passkey, then the passkey, for everything else.
Passkeys work well, but I feel as if they are still a bit too “fiddly” for your average mensch.
I’m pretty sure the private key is transferable, but Apple keeps them in the Keychain, so they seem locked into the OS. On my machine, I can also keep passkeys in 1Password.
A while back, I wrote up a series on implementing passkeys in iOS/MacOS: https://littlegreenviper.com/series/passkeys/