I think you are overthinking this way too much. A right to roam would have no impact on road building or other public projects.
Well, in Scotland you only have the right to walk or cycle across other people's property - and with a number of restrictions (no damaging crops, no access to immediate vicinity or houses etc.).
When I was in Scotland two years this was a breath of fresh air and “freedom”. In the US there are lots of areas where it’s really hard to hike because people close off their land completely. Or they mark the land as private and now you have to worry about getting shot or somethings.
If anyone can just go through the land anyway, all of a sudden there is no need for any projects. At least no big one.
But maybe I'm just being cynical.