Other points to bear in mind are, with drip (V60, Mugen, Kalita, Melitta) you can hit high extractions in 3-4min, the equivalent takes 10x longer with immersion.
The other point is that in drip & Clever, the grounds are part of the filtration system, as well as the paper. So if you pour a French press into a V60 paper you still won’t get a brew as clean as a V60…I never much liked the change in taste in doing this either, the result was always a bit harsh, I found Filtropa white papers better & Chemex white papers (in the end it is easier to be careful with a French press and get clean enough brews, without paper filtering).
Also, filtering short FP brews can make the under-extraction more noticeable, when the non dissolved solids are filtered out.
Also, as there’s no bed in a bare filter when you pour a brew into it, you see quite a bit of brew hanging around in there for ages, draw down takes 10s of minutes sometimes.
FWIW I wasted months trying to make filtered FP brews in the office, it’s basically a bad idea.
