I understand. I guess this could be done using gravity alone, but would take a very long time. Seems like some kind of pump or pressure system would be needed. I have no budget so just dreaming really.
I agree, the finer the filter, the longer it will take to filter. That, combined with the viscosity of the chosen solvent (PG, ethanol, water, VG, etc.), will influence how easily/quickly a solution can be filtered. Each DIYer has to choose what level of convenience and quality one wants, and what they are willing to pay for it.
The AeroPress is basically the same thing as a giant syringe. I have read accounts, of users placing weights (books, gym weights, etc.) on the plunger, so they can go about other activities while the press does its thing.
Please keep in mind; I do NOT do any type of extractions myself (I am perfectly satisfied with the plethora of available concentrates). So, for my part, this is only regurgitating what I have read on the
tobacco extractions sub-forum I previously linked. Check them out, and you can ask practical questions there, for a more informed/experienced response.