I have a french press for camping, it's fine but when I want that first cup of coffee and it's cold and rainy outside you better believe I'm not pressing it slowly.

Also my reaction the first time someone tried to tell me to plunge slower.

>Have to imagine the coarseness/evenness of your grind is a 10x bigger factor in that regard

100%. Plunge speed makes a minor difference at most, and only then when you have a suitably coarse and even grind.

If you want to play with the concept I think the best way is to prep a press, but instead of plunging the filter into the coffee before pouring, press it only to the top of the liquid and pour, allowing gravity to pull coffee through the filter. Pour a small cup like this, then press the rest of the way and pour another cup.

The original suggestion of plunging slowly and gently is entirely sensible, since it's the only variable available in the context.