In an effort to lower the temperature on the topic of National Forests and legal access, the best course of action is to get the "Motor Vehicle Travel Map" that I believe every national forest has nowadays. That way, you will have a legal document that clearly shows you where the public roads and trails are on the forest, as well as showing public/private land ownership. Although mapping apps will generally have this information on them, as well, they are not always accurate and could get you in trouble. Get the travel map!
I suspect there are private landowners across the country within National Forests that will do what they can to limit access to forest lands, including locking up legally-established, public roads (at least that is the case out here in the west). Conversely, there are also roads on NF's that cross private property and then go onto forest land that are NOT legally accessible. The travel maps will clarify any/all of this for you...and if you find a road that is locked up that is a designated public road, you have a government document to support any effort you might make to gain access.
However, it is advisable to not take matters into your own hands when faced with an illegally-locked gate. Report any such incident to the governing authorities and have them deal with it...if you have the time and patience to wait on a resolution.
