I’m not an expert in the mechanics but I think there is also a way for private credit creation to run as a separate operation, similar to how loans were originated in the sub prime mortgage crisis and then immediately sold on. I’m not recommending that happen, but a similar system would allow private credit creation without the loan issuers being the primary deposit taking institution.

But yes private banks could also operate term deposits and pay interest to attract deposits to satisfy net flows of funds with other institutions.