Some banks do, some don't.
In a lot of countries if you establish a small company and open a bank account for it they will just do the bare minimum, after all if you got a tax number and just got a new company registered you already jumped through a few background check hoops.
Of course they will still run your name through their whatever systems. And eventually if your business starts transacting it'll be constantly looked at by the automated systems.
As long as you don't do anything suspicious, no one will care.
Basically if you behave like all the small shops that have to deposit cash, no one will look twice.
Also, cash is not interesting, the problem is when you start to wire money (or get transfers) to/from places that small kitchens usually don't. And that's when risk and compliance departments might notice the activity.