Plywood - and edge banding - is not nearly as durable as solid wood edged and veneered doors/drawer fronts. Problem is you simply can't get the solid wood edging "under' the veneer yourself without ... veneering. Plus, plywood panels from big box will undoubtedly have limited grain, and are typically rotary cut.
Possibly consider having a specialty firm do your veneering for you? You can solid wood edge the mdf and deliver and they will prep and veneer, or you can provide the spec and they will fabricate. They can use a 1/16th" veneer which will be substantially easier to work with and more durable in use. The guys I've used (Cairo, NY) are fantastic and much faster and more consistent than I am at veneering. That is actually my preferred approach if building more than a small vanity like I show above. For the end panels perhaps your design approach doesn't require solid wood edging pre-applied.
Same idea as drawers. While I can build drawers (and actually did for a job this weekend) the rational approach is to buy them. I have a local supplier that specializes in drawers and cranks them out like hot dogs!
"the mechanic that would perfect his work must first sharpen his tools.” Confucius