For online payments, is it better to do it through my bank instead of using the payee's online payment system?

When making payments such as utility or insurance bills, I’ve always used the payee’s systems. I’ve never used my bank’s online payment system. I wonder if that is a mistake.

I may be putting some stuff in storage while I go abroad. I want to have a way to pay my monthly storage fee remotely, and the storage places I’m considering don’t all have their own systems for that. I was looking at setting up payments using my bank’s online payment system, which asks for the payee’s account number.

It occurred to me that maybe I should do that for everything, because it would be more secure (for me) to give someone else’s account number to my bank than to give my bank account or credit card number to someone else.