Exactly. BITD when I owned a brokerage I'd write up an offer for a cash purchase and then the buyer would go ahead and get a mortgage if the buyer wanted to keep their cash. Nothing stopping them from doing this, the only difference is that the buyer didn't have the mtg contingency clause that would allow them to walk if the mtg wasn't approved.
They had the cash and could close the deal but they chose to get a mtg. Some listing agents seemed pssd because they, again, were so stupid that they couldn't understand the concept. (Yes, I have a fairly low opinion of the average agent. May be a lack of training or simply a lack of IQ, or both. In either case, most should not be involved in the largest financial transaction most people will make in their lifetimes.)
All a cash offer does is remove the financing contingency; its really no business of the seller where the buyer gets the cash, from a checkbook or from a lender.