So is the Cumberland Road substantially different than the public works projects of the New Deal
Yep. the need for the federal government to create those pathways of commerce to unite our new republic was especially critical in the time of Washington and Jefferson.
So are you backing off the "military road" falsehood?
I've agreed that by your standard - the requirement of an official declaration of it's status as such, you would consider that characterization a falsehood.
If they are fundamentally different then one cannot be held to be a valid precedent for the other. The abuse of precedent, particularly when compounded by building precedent upon precedent is exactly what Jefferson was warning against in his letter.
We have the assertion that this set the precedent for all future federal public works, in the context of being in response to his warning about such abuse.
If we can assume that it's possible that use of precedent may be flawed, then the obvious differences in this and the make work programs indicate that we have arrived there by the very process he's warned against.
If we must assume that the use of this as precedent was legitimate until proven otherwise, then in this context it appears to be an attempt to paint Jefferson as the "father of all federal public works projects", and a hypocrite for not following his own advice.