I used mdf core cherry plywood for a kitchen island recently. The 1/4 thickness worked well but in this case I was not using a cope and stick bit so I could have adjusted the thickness of the dado.
I have used so called 1/4 veneered plywood before with cope and stick bits but it does leave a gap. Finish may fill it but there is still a gap. On painted doors, I caulk it before finishing. On clear, I leave it and let the finish stick the panel in place. I've done this on oak which to me is not a real pristine kind of wood - more character than fussy.
I normally do not deliberately glue the panel in place when clear finishing because that tends to get glue on the panel. It makes the door stronger, however, and you do not worry about wood movement with plywood (at least I do not). I always finish solid wood panels prior to glue-up, however. I have seen unfinished edges and do not want any on my work.
If I wanted to minimize the gaps on the side of the door that shows the most, I would wedge it from the back prior to appication of the first coat or two of finish. The finish should stick it in this position. That maximizes the gap on the other side, however. If you want to get real tricky, you could cut a little shim and drop it in the groove to close the gap. I do not find the gap to be that big of a deal, however.
Jim