I already have this problem; both my bank and my credit union introduced 2FA but only with SMS. Once enabled, any attempt to log in using a not-yet-authorized browser or app is stalled until I get that text message. Presumably a call to customer service would sort it out eventually, but that prospect isn't terribly pleasant.

Every time I have to call customer support to reset a bank password it makes me realize how bad of a security hole most phone support is. Security through two-factor authentication is only as strong as the process for bypassing it.