Password and 2FA

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I am not an expert on this at all, but I think being able to sync 2 factor authentication data with a 3rd party provider and access it through the web would defeat the purpose of 2FA.

But maybe I am misinterpreting what you mean ^^"

Sorry if you already know this, but since you mentioned the issue of losing your device, I thought maybe you didn't ^^"

If you do lose or otherwise make your 2FA device unavailable there are usually security measures to get back into a good state. It could be through a unique recovery key, you usually can get a copy of from the place you enabled 2FA, but it can also be some secondary device being sent a recovery code. Some services will let you access the recovery keys several times, others will only allow you to copy it during initial setup.