The texture will come from the roller. That is the down side to those foam gloss rollers. The finish is all about feathering. IE, get the paint on, then roll it over using 0 pressure. That will smooth it out, with sanding between coats. Painting raw MDF is not easy. If it kicks your ass I would just order some satin vinyl and wrap the bugger. Very hard wearing too, it's how I used to refurb jamma cabinets BITD.
TBH my first thought was that it was user error, so clearly I wasn't wrong A vinyl wrap is not a bad idea. Regardless of what I do with the finished product, it'll still need sealing.