What thickness of veneer did everyone use?
Good question.
Once upon a time, veneer thickness (or thinness) was more or less constrained by the machinery of the time. These days, veneers are also offered in very thin forms and some are paper backed. Worth noting that much of todays production is aimed at markets consuming large quantities of veneer for the manufacture of panels using the very uniform 'engineered boards' as the substrate (like MDF).
For the home gamer doing a dashboard, there won't be much room for sanding if using 1/64" paper backed veneers. This is a worthwhile question for the veneer supplier as there's likely very few people using traditional hide glue. Remember too that the interior of a Europa on a hot day is very different from the environment envisaged for Aunt Edna's chiffonier.
But I reckon, with some forethought, one could get away with a paper backed veneer provided you planed the ply flat and I recall Richard is proposing to vacuum bag the dash, so that all reduces the risk factors.
Pic of Aunt Edna attached for reference.