There is, in a theoretical sense. Possibly not in a practical sense.
A recreational herd/fishery is very different from a commercial one. The scale is smaller. The self interested political incentives are usually overshadowed by social/ecological/altruistic interests. There are no "commercial viability" concerns, so all sorts of arbitrary restrictions aren't a problem. The incentive to cheat is much smaller. Recreational hunters and fisherman always make arbitrary rules about what's "sporting" anyway. Fly fishing only. No baiting... These are usually ways of lowering productivity, not increasing it. The whole thing dovetails nicely with ecosystem preservation, etc.
Commercially, the US decided a long time ago that deer and such should not be commercial commodities. Did they leave some sustainable harvest potential the table? Sure. How big was it? Not big^. But, they took away huge risks of ecological harm.
Same question for ocean fisheries. Are there some theoretically sustainable methods/quotas/rules? Sure. How well will they be written & policed in an international, commercial environment? Questionable? How important is commercial fishing of wild populations to us? Not very, and less important every year.
^Obviously, this is relative. In the context preindustrial economies and their economic output, deer (or trout, etc.) may have been very important. In the 1800s, less so. In the 21s century, the potential contribution of wild deer to the economy is miniscule. Just like you say, recreational value is probably more valuable and than the actual outputs (meat, hides, etc.) even in money terms.