I did a search for you and typed in 'dark wainscoting'. Look at all the examples that came up :-)
https://www.houzz.com/photos/dark-wainscoting-phbr0lbl-bl~l_44794

I have painted beadboard a medium gray green for a client and it looked great (sorry I dont have pictures of this project).
PyramidDG.com

Style: eclectic cottage. Decor is silvery/taupe faux shantung draperies, oak pedestal table and plain 70 yr. old oak buffett (small), chairs to be decided on, composite (?) wood floors to go in w/6" baseboards. Art is B/W & sepia photos in black frames.

The whole wall, floor to ceiling, is beadboard. (I took 4' off the dining room to create a storage room so this is a "new" wall.) I'm leaning toward one or 2 shades darker taupe than the draperies. The floating shelf will go across the entire beadboard wall, ideally painted a vivid turquoise, but that 's flexible depending on wall color. Arched doorway into LR which is light golden tan so I want it to blend.

Would flat paint be better than than eggshell/satin finish?

I think the beadboard should be painted satin, brushed smooth not sprayed.

Even semi gloss.

When you say a dark color…black, navy, brown? with the turquoise floating shelf running across? I bet you could, but you need to highlight the dark.