The hunting mechanic has been fixed in the Vikings add-on for Settlers 2 10th Anniversary
I have just had a long (10+ hours) of game in Settlers 2 10th Anniversary through its Vikings add-on, and I want to report my findings regarding the hunting mechanics of the game: the original issue of the hunter becoming useless after the handful of nearby animals being hunt down has been fixed. Animals will never run out, if you have a sufficiently large forest next to the hunter shed. I even tested the size of the forest, and the larger the forest the higher the hunter's efficiency will be. Visually you won't see any animals in the forest, but the hunter will keep going out, go under a tree, and emerge with meet in its hands.
What I haven't tested yet is if this mechanic would work with planted forests too (through the forester) completely from the ground up, or in areas where there were no animals to begin with. It definitely works with small wooded areas where you use a forester to extend the size of the tree covered area.
I also just want to mention I have only had one play session so far on a user created map (I think it's called Beautiful World) where I actually noticed this mechanic, as normally I barely ever use the hunter, but I don't see why it wouldn't work the same for every other map in the game. Either way, if it turns out I was wrong, I will report back here.
The reason in the base game I barely ever used the hunters was that they became useless very quickly after the handful of animals were hunt down, but with this new mechanic it's now actually a very useful building: immediate food source without needing to spend a ton on infrastructure: for example for bread you need multiple really large and expensive buildings. For the hunter it's literally just the hunter building and a forest next to it.
The important part is you need to play the game through its Vikings add-on (there is an Englisht translated version out there), there in its menu you can play both the original campaign or the new Vikings campaign, and all the other fixes (like new military outposts having no gold coins sent to them by default) will be applied to the game too. Seriously this is the definitive way to play the game.