Hi there and welcome!
Well if you follow the USDA guidelines for ground beef or pork is to cook to 160F internal temp or higher.
The USDA guideline for whole muscle pork and whole muscle venison is 145F internal temp or higher.
So cooking venison burger or any of the other ground meat to rare is technically below guidelines. For myself, If I were to do it "rarer" than recommended guidelines I would still try to at least hit the whole muscle temp... though its not an apples to apples guideline and comparison.
I mix ground venison with either-or beef fat or pork fat and make burger with it. I grill at a hot temp to get that good sear but I don't cook the crap out of them as venison cooks up fast. Even with my regular beef burgers get a good sear and don't cook too hard either but rarely is there much if any pink them.
Now most of my life I process my own venison so I know it goes from whole muscle to grind within hours of time and then into ground meat bags or vac seal bags within 24 hours but refrigerated well during the whole process when not being worked.
So I would have no problem cooking my own ground venison to a "rarer" temp but it still feel like no matter what, you throw a venison patty on a grill and look away for 2 seconds to look back and find it about cooked lol :D
The main thing is bee safe with your food so you enjoy it and don't end up sick and still make something tasty. I hope this info helps some
I used to be the food safety "police" registered dietitian at nursing homes and hospitals. for many years. No way would I let them do that! But, for us, I want mine a little pink. We do our own game processing and we're very particular. (My wife fails to understand that well-done is essentially a sin! LOL) I would not cook burger I buy from the store to less than the recommended minimum temperature There is too much risk that it was exposed to e-coli, etc. And, believe me, it is often not handled well.
Here's something to think about, though. People might be really careful with internal temps, but not so careful with cooking utensils and dishes. For example, if you put a raw patty on the grill with a spatula, or you sure the spatula got hot enough to kill bacteria from the raw juices when you take it off the grill? Possibly. But if someone puts a cooked burger on the same plate the raw meat was brought out with, it's a big risk.
My wife and I killed two cow elk at once this year in late August. I had 350 pounds of homemade ice in Ziplocs with me and knew I would drive up close to the kill to retrieve it. I had already asked someone to help us with the retrieval and field dressing in exchange for one of the elk for him, if that occurred. I explained how we needed to ice pack it quickly in sleds in the trucks.
He refused to even skin and quarter his in that warm weather. I was very careful and controlling with our elk. I watched him lay his meat directly on the bed of his truck that was caked in blood with his footprints in the blood. He also used the same knife for dirty jobs and cutting off meat when we hung it and skinned it on my meat pole at my home. The whole time he was showing me "tricks" he learned when he was trained by a butcher. I already knew those "tricks" from 40 years of processing my own game. I felt like telling him, here's a trick for you. "Cook the hell out of your meat, since it's definitely contaminated!"
The contamination is the unknown factor sometimes.