Looks like you could put a car on that standVery well built.
Yep putty up those screws and go flat black, I used gloss and really regret it. Imperfections you couldn't even see before the paint really stick out. Painting is a smelly, messy hassle and almost impossible to get a perfect finish, well for me anyway.
If I could go back I would krylon fusion (spraypaint) in flat black or satin.
looks great!
Leaving it unfinished is fine, but you want to put something on there to protect the wood from the drips and everything else that will happen. You especially want to seal the MDF. If you don't seal that, you'll drip some water on it and ruin your doors.
Just looking out for you and wouldn't want a nice stand like that to get messed up.