ehm…no, this is not correct.
Bootloader unlocking and rooting are different procedures: you can have an unlocked bootloader without having a rooted system.
It depends on how did you root the phone… If you used Magisk, you can enable the MagiskHide features, that allows you to chose which apps shouldn’t see that the phone is rooted.
Some home-banking apps can also check if rooting applications are installed: Magisk allows you to install its managing app (Magisk Manager) with a different name, so it doesn’t get recognized by other apps.