What payment method would you recommend for paying mortgages, utility bills, and company expenses?

You said you're in the EU right ? SEPA works. The only moment I need to confirm identity with my bank is when I buy frivolities, not for my bills.

Here's the problem: I can do a SEPA transaction using online banking (with a Google-enabled phone) or by walking to my bank personally.

I might be lucky enough to have online banking work with sms and a pre-shared secret. My bank (also in the EU) didn't tell me I could do it automatically, they only mentioned it when I told them I lost my phone (it was true) and that I didn't know when I could buy a new one. Maybe it'll work with yours ?

No, ever since PSD2 came into effect, banks here refuse to do SMS-based verification and have switched to apps. They also don't support hardware authenticators for consumers. I asked.

Fortunately we have competition and possibility to switch to banks that are more private and user friendly.