Every QT tank needs a filter. If the goal is to prepare a QT tank, it must be seeded by something containing nitrifying bacteria. You said water from a fully established DT (20 gallons) will not have a sufficient amount of this bacteria to handle the bioload of a QT tank because there is not enough free-swimming bacteria. This in incorrect. My QT tank holds 15-20 gallons and all of that water comes from my DT when I set it up. Freshly mixed saltwater goes into my DT to replace the water removed to start a QT. The water from my DT will seed my HOB filter and the marco rock in my my QT.
I think you are misunderstanding me. I'm not saying a QT tank with nothing but DT water is enough to perform the biological filtration. The bacteria will need surface area to colonize and perform that function, whether it is a sponge or live rock. No QT tank should ONLY have water in it, and nothing else.