This is all I know at the moment (our president, who knows all, is out of town)--our coop has an account at the bank. My accounts are with Chase, and I set up autopays for most of my bills, including maintenance, so I don't have to remember to make payments every month.
It does not matter where any of these payments go, when I set up "payees" in the account, I input the bank name, address, and our account number, where the maintenance payment goes. It does not have to be paid into a Chase account because I am paying from a Chase account. I don't recall doing anything more than that (though I might not be remembering--I started this 6 years ago when we moved here.) When I have to make additional payments to the coop like laundry fees, which we pay quarterly, I go into my "payees" list in Chase online, choose the coop, and just enter the payment and press send.
I choose when the bank is to send the payment when I set it up, say, on the 21st of each month for the next month. The system tells me when to send the payment so that it will reach the coop account by the first of the month. Chase actually sends a printed check, so you have to send out the payments about 5-7 days before the due date.
Sometimes there might be a glitch in the system. I think one of us once had her payments somehow get lost in the system and not reach the coop account--and that had to be sorted out. We have a couple of elderly tenants who may not be using this system--at least one of them does not use a computer-- and may actually visit the bank and deposit a check into the coop account. I really have no idea, but I am pretty sure that nobody is sending checks to our treasurer for him to deposit.
I do recall there being a bit of an issue because HSBC has no way of noting where each payment came from, so it's not so easy to figure out who did not pay if there is a shortfall in the account. Once there were missing payments, and we all got an email from our treasurer asking us to make sure we had paid every month, and to document that. It turned out that one person's bank had stopped sending her autopayments, because they had a rule that autopayments had to be confirmed every six months. She did not realize she was behind on payments.