I’d rather not switch to authenticator if I don’t have to. I’ll try your suggestion of reaching out to my service provider and see if that has any positive results.
If you don't want to use authenticator because you are boycotting Google products, you can use FreeOTP which is open-source and works fine.
This is their website if you want to find more infos about it :
https://freeotp.github.io/
Otherwise, philip's answer could be a good solution for you. Playing with the autowithdrawal function is a very good way to avoid SMSes, you can turn it on only when you want to withdraw, and without SMS verification.
But with this solution you will need to be able to receive their SMS at least one time to activate the service.
Yeah When the account starts to have a decent amount I turn on the autowithdrawal