a rent/letting agency says they ''cannot'' accept payment by credit/debit card and want to be paid by BACS/Online payments ... why?

Not shady at all, all it is is a standing order, the money comes out of your account to them each month.

Employers do BACS, they don't hand over cash for your work, they pay through your bank.

This is normal practice in the U.K.

they don't accept credit cards, that simple

Credit/debit cards charge those accepting them a fee. Plus your LL has to set up ability to accept credit/debit cards with their bank & that costs money, too. Besides, a small LL will not qualify with the bank to accept such cards - they don't do enough business to meet requirements.

Online payments go bank to bank without fees to recipient. Usually no fees to sender, either.

Credit card payment incur extra charges for them usually 1.5% I wouldn't take them either for something as large as rent

Because if you pay by credit card, the agency has to pay a certain percentage back to the credit card company. That’s why most Chinese restaurants don’t accept credit cards. They want cash so they don’t have to pay any fees to the credit card company. It’s also for paying the workers under the table.

It could be because banks charge them for such payments as a processing fee. It is relatively common fora bank to charge 2.5-3% processing fee for such transactions. This would cut into the profits of a business. It may seem like a small amount but it comes directly from the profits of a business.