I'm always afraid of my phone getting stolen or losing it somewhere so I have a completely separate iPhone, which runs my banking apps. I keep that phone at home.
Depending on where you live, a burglary might be more common than a robbery. Why don't you just use the bank's website on your desktop computer (assuming you have a desktop computer)?
Because in infinite banking sector's wisdom, logging into the website requires a confirmation with the mobile app.
I've changed banks for less.
I'm in a similar position and I hate it. They somehow managed to convince themselves that if you issue tokens for 2FA within the mobile app it is still "two" factor authentication. Of course since you already have mobile app now, you can just use it directly (and there is no way to disbale that). So while webapp is 2FA, there is now a mobile app which is not. Good thinking.
Are there banks without such requirement these days?
Schwab works with totp as 2fa.
Last time I looked they required some Symantec BS to intermediate. Has that ended?
I don't know if it has ended but you could reverse engineer the Symantec BS and convert it to regular totp. You likely need root to extract the internal store from the symantec app.
Ask for an alternative authentication mechanism because you "do not own a smartphone".
All banks I know that do this have an alternative 2FA you can use instead tho.
I'm concerned about losing phones too, so I don't bank on any phone.