I can give you the answer, and if applied you'd get a really nice smooth finish. Up to you.

That type of board presents a surface of varying densities, there is only one solution, cover it and seal it completely, either with laminate or high build (sprayed on) primer and paint.
You could roll on a thick primer, but it will be a PITA to get smooth for painting. Spraying 3 or 4 coats of high build primer and sanding between each until all signs of underlying texture are gone is the way to go. Then spray paint after that.
High build primer in a rattle can from and auto paints supplier. It is not the same as just primer, it contains a filler putty as well.
I wouldn't start over with MDF, it presents similar issues, just not as much as that type of board, you still need to prime it well if you are going to paint it or the result is going to end up rough as well. Laminate is a great alternative but comes with its own challenges.
I forgot to mention, a machine sander (hand held mechanical sander) would be very useful in getting a nice even surface if using a high build primer.