The only time I’ve had a gut shot deer affect the meat has been as a result of adrenaline affecting the situation, not the gut spillage itself. If the deer ran a good ways, the gut shot protocol wasn’t followed (back out eight or so hours later, let em bed down, etc) then adrenaline becomes a factor, and you are dealing with some potentially tough meat.
Some of the toughest meat I’ve ever encountered has been from dog drives and the deer are running for their lives. Like dealing with gut spillage in the meat, I clean everything out, ensure the carcass goes through rigor mortis, age for as long as I can and then process. 95% of the time you can’t tell any difference from cleanly taken deer.
