This is not correct. Look at any county or other map that provides designaltions on the type of road. You will see either "***" or "FAS" on virtually all roads outside city streets and the Interstates.
Not up here in rural New York you won't. There are a vast number of NON-federal-aid roads here. Federal-aid-eligible roads are a pretty random mishmash of roads, certainly.
The federal aid eligibility is apparently defined by the idiotic "highway functional classification standards", which assume a "tree and branch" road system, which of course is completely bogus.
This classification system assumes that there are "local roads", "minor collector roads", "major collector roads", "minor arterial roads", and "principal arterial roads", and tries to shoehorn the existing roads into this system. Into which they do not fit.
Apparently "urban principal arterial" through "urban collector" are federal-aid eligible, while "rural principal arterial" through "rural major collector" are federal-aid eligible
This means that a perfectly random set of roads are eligible for federal funding. It tends to include most of the state numbered highways and some random other roads. But MOST roads are not eligible for federal aid.
It is further worth noting that it is often necessary to *refuse* federal aid for eligible roads. Why? Because the roads may actually be local roads with houses on both sides and "children at play" -- but the federal aid guidelines are likely to want to construct them like speedways, due to this "functional classification" garbage.
The rich suburb I grew up in had to reject both federal and state money for several road rebuildings, due to these "widen the roads make the cars faster" requirements attached to the funding.