If your joints are clean and flush, glue joints will be the strongest part. Using a good wood glue such as Titebond and a sufficient amount of clamps, you will have a cabinet you can work with within an hour. I would recommend allowing the glue to set with the clamps for a least a couple hours though. Overnight if you have the patience.
If your joints are not flush, countersinking a few screws to hold the outer shell together won't hurt anything. Not needed for the braces as they have plenty of surface area that will see glue. The less screws you have to use, the less filling you will have to do.
Just don't think that by using screws you will save time by not having to wait as long for the glue to set. Filling in the screw holes will eat up more of your time. At least you can be doing something else while the glue is setting.