I'm going to post a counter-opinion and say it does matter. Not because that fake structure makes any difference to the sound of the piano - it doesn't. But...
We do not need it to be mobile
I'm not sure that's necessarily a good call. If your young student is going to grow up and might want to take the piano with them, a digital piano with all the fake woodwork is not very practical to move around. And if your reason for buying a digital piano instead of a real one is to save space in your house, it's way more convenient if you can fold the piano down and stash it under a bed if you've got guests.
A stage piano without all the fake woodwork bolted to it, and with a good collapsible stand and collapsible piano stool, is likely to be a better call generally in both those cases. By all means leave it assembled and ready to play, most of the time. But it gives you options which the "cabinet" one doesn't.
For a young student, a stage piano also has a massive advantage ergonomically. The piano and stool can be lowered to their height - they don't have to perch on a too-high stool with their feet dangling in mid-air. Don't underestimate the impact of simple physical discomfort on how much someone wants to practise.
And lastly of course, with a stage piano you're only paying for the actual instrument. With digital pianos in cabinets, a significant chunk of your money is paying for the cabinet. If you're looking for a musical instrument and not a piece of furniture, spend the money on the musical instrument.