So here is a little background as a land manager for one of those federal agencies. I won't say which, and this is in no way an official opinion of the government or my agency. These are my observations having worked for 3 of the natural resource agencies (2 land based, and one wildlife based). This is the extent of our involvement in the hunting world, I will also cover a few reasons why fees and laws are changing in Idaho.
1. We collect zero revenue from the hunting community in the form of tag/license fees. The extent of federal revenue collection from hunting comes from camp site fees and a few cents from your stamp (most of the revenue goes to the state or the treasury and never goes to the agency). I won't speak to the park service since I haven't ever worked for them, and generally avoid national parks.
2. Other than USFWS we have almost zero say in seasons, tags, units, or how states manage game species. This is to the point that in many ways federal land agencies (other than USFWS) can't even comment on state policies or rules. USFWS can make some comments on certain species, but that is limited to threatened and endangered species, and potential impacts to them (i.e. salmon, sage grouse, etc.)
3. We will regulate OHV or shooting use of public lands. Usually trail and shooting area closures happen because of how people are using the resource. That is very unpopular to many people, but the sad truth is that more people than you realize are damaging the lands. In one year I have found thousands of pounds of garbage, used needle dumps, wire/copper burning pits, asbestos dumps, and other things that would make you sick. These items are getting dumped all over our public lands, and after we blow half our budget doing hazmat cleanups every year, it gets hard not to consider closing areas. Trails are for multiple reasons.. It could be because we have active logging going on, and we don't want a user to get run over by a deck truck. Most of the time it is because people are riding off trail, killing some endangered plant, dumping gas, oil, or antifreeze in the middle of a creek (seen that 4 times now), or just causing a danger to the community (in north Idaho this is usually because of a multi year increase in OHV injury wrecks and drunk/intoxicated driving. If 1 in 3 people drove on the interstate drunk all day long from 0900-1900, would you be surprised if they closed the interstate for a while?).
4. We will temporarily close some permitted activities during hunting season, or limit their operations to increase public safety. This is usually in the form of temporary logging closures, or like funtions.
5. We will increase our law enforcement presence to make sure that folks aren't off trail with their ATVs, aren't driving drunk, aren't squating on the land, and basically just increasing for 5 day to 7 day a week coverage during the week. This also provides support to Game Wardens because they have limited backup as it is.
6. In very few cases, areas have been closed to hunting due to public safety (a recent lodgepole burn where snags are falling every few minutes) or proximity to infrastructure that has been repeatedly damaged by hunters or shooters.
I'm sure I missed something, but in the end the agencies of Uncle Sam tend to treat hunting as any other recreational activity. So in the end, yes they are your public lands but those fish and game resources are managed by the state and we have little to no say in how it's done.
As far as Idaho goes, and the recent change to license/tag fees and a few law changes have all been due to resident public comment and simple market analysis. Idaho had some of the cheapest nonresident tag fees in the western US, and it was being used as a second choice location for people when they didn't draw elsewhere. Idaho realized that they could increase cost of nonresident hunting tags by 50% or more and still be $200 cheaper than many of our neighboring states. People are saying that Idaho will lose money in this deal, and that they wont come here to hunt. Honestly that's not going to happen, it is still cheap enough that the tags will sell out like they have for years. People also claim that residents are buying all the nonresident tags as second tags. This is false. Idaho doesn't even allow Idahoans to buy a second tag until August 1st, while nonresident tags go on sale december 1st of the previous year. That is 8 months of priority over residents getting a second tag, so please don't say that is limiting your ability to get a tag... In 2018 and 2019 nonresident elk tags sold out before Idaho residents could purchase a second tag, and deer tags have been selling out by the end of September since 2017. They also realized that many people were illegally claiming residency in Idaho when they lived elsewhere to take advantage of resident tags, and more so the allocations in the controlled hunt draw system. This has been a massive issue in Idaho, and that is why Idaho changed its residency requirements a while back to say that you had to be a sole resident of Idaho and nowhere else to receive resident hunting privileges. I have been on multiple hunts where we were one of the only Idaho license plates present, but most of the other hunters were claiming residency because they had a bill sent to a friend's house or they rented a studio apartment with a bunch of their friends. That isn't an unsupported claim, that is direct from the other hunters as they bragged about how they played the system... Boy were they surprised when my hunting partner informed then that he is a Conservation Officer (game Warden). HA!
In Idaho the fish and game is funded predominantly from licenses/tags due to an unfriendly political system in Boise that dates back to the Butch Otter days. And as such, I personally wish that they would have doubled both residents and nonresidents alike, I am more than happy to support IDFG or any other state game management agency. Those folks work incredibly hard, and for very little pay, they deserve the extra money. Conservation Officers in Idaho start at $18 an hour and requires at minimum a 4 year degree, biologists start at 18-20 and lately have been requiring a masters degree and multiple years experience. In the end both will be lucky to make $24 an hour, and are sacrificing allot to provide us with the opportunity to hunt and fish. Do I agree with all of their policies? No. But the vast majority of their work is really an awesome resource for the rest of us.
Overall that is a short novel on how federal agencies have very little say or influence on hunting, and why tag prices are going up in the state of Idaho. Thanks for reading, and feel free to ask questions. I don't have much better to do while I heal up, so if I can't answer, I can probably find someone that can.