>What happens when I go to the same site on a different platform?

Scan the QR code with your phone.

>What if I can't use cross-device authentication because I'm on a locked-down work machine or one that simply doesn't have bluetooth?

Most devices today ship with Bluetooth and even locked down machines support using your personal phone to authenticate.

>What if I want to consolidate them all in one platform just for the sake of being organised?

Then do that. Use whatever your phone uses.

>What if I lose or simply don't have the device with a certain passkey near me?

Then you will need to use a fallback authentication mechanism or will need to wait until you can get the device. TOTPs have the same issue.