3/4 plywood over a 3' span is likely to be fine as long as she does not sit or climb up on it, or leave heavy objects near the edge for long periods of time.
I am sitting at a desk, as i right this that is 3/4 plywood spanning 6' and it is fine. If i put a straight edge on it i think i may see a sag of about an 1/8", with out my heavy arms resting on it to use the keyboard, but it will flex significantly if press down on it.
I think it will be fine but since you have not built it yet you may want to go with either a thicker ply, thicker solid wood or butcher block. A
Aesthetically i think it will look better on the thick base.
And would the answer to this be different if I edge-banded the plywood
with 1 inch of hardwood (walnut)?
The thicker walnut hardwood would add stiffness but not take all of the flex out of it.