I'm skeptical of 2fa, since SMS for example is not a secure way of transmitting information. Nowadays, the underlying encryption can be hacked quite easily. The point of 2fa is almost completely erased, if you use the same device for opening a web wallet and receiving SMS. In addition, by using 2fa, you reveal your phone number to the web wallet operator which has negative privacy and security implications. For example, if the wallet operator is hacked, your phone number might be revealed to criminals and you could be an easy target of personalized scam attempts.
So if you want to use 2fa, better use a specialized device with a local wallet.
For storing larger amounts of Bitcoin, I would recommend cold storage only (for example paper wallets). This way (if the addresses were generated in a secure environment) you eliminate the risk of hacking and third party scams altogether.
ya.ya.yo!
I have had a problem with 2fa as well.
It was from my Google account. I had switched sim card since I needed a new one to refresh my phone identity and unfortunately, I lost my old one.
I did not really think of it much until I had to log in my account again from a different pc.