Not sure if it flags as fake but I'd look into getting a dedicated Twilio number, then just forward incoming texts to your email or something like that. It would at least get the "shady" part out of the equation as Twilio is pretty trustworthy.
This does not work, I've tried this before. Google verification for example would not accept my Twilio number as verification (about 2 years ago). You can lookup a phone number for the provider and numbers from Twilio or others tend to not be accepted.
I'd consider them more trust worthy to pass me a MFA code than some random shady website that the GP was currently using, but sounds like people have tried this and Twilio numbers are on a ban list for many services
