I volunteer at a place that also functions as a food bank for people in need. You should see some of the people that come through. I don't work in the food bank portion but I do give free eggs and veggies. You might get some people that don't eat certain things but I assure you many people will be grateful for whatever they can get. Frankly food banks not taking venison because the people would not want it sounds a lot more like an urban legend than anything else. As the guy running the food bank said, they will take anything because someone will eat it. I've given them some weird things I've had to tell them how to cook like Tromboncini squash and they always give it away.
It is possible state, city, or county health department rules don't allow them to take venison, especially road kill. In some places by law road kill goes to orphanages or some similar place. Maybe by law it has to be processed a certain way before they can accept it. I can easily see food banks not able to accept whole carcasses or unprocessed meat, but processed meat, especially if it is individually wrapped! Not for that reason.
Some places, like Goodwill, have rules about what they will accept and what they will not. I've been surprised by some of those. Some food banks have rules about what they will not accept, like canned goods out of date or with labels ripped off. Maybe they don't have refrigerated storage facilities. I used to take Boy Scouts to one of those so they could sort food and get their service hours.
There are many possible reasons a food bank might not accept certain things and you may have some people with certain beliefs running them that impose their ideas, but a food bank not accepting food because no one would want that just does not sound true from what I have seen. It is going to be a law, some logistics reason, or the ideosyncracy of the person running it.
It takes someone to set it up and the law has to allow it, but harvested venison can go to schools, prisons, food banks, orphanages, or other legitimate places if someone will sort it out. That meat has to be handled right and butchered, but it can be used by someone.