I'm pretty sure that this is an old wives tale (no reference to your Mominlaw badshotbob) that has simply been allowed to stick around.

I've had meat from deer that have been shot on the run, shot multiple times (shotgun hunting with a gang) as well as deer I've dropped with one shot while they were calm and still. I haven't been able to detect anything different in the taste and I'm always conscious of the flavor as my kids are finnicky about the taste of food.

The real key to good tasting meat is deboning it before butchering and leaving as little tallow, silverskin and sinew in the cuts as possible. I try to minimize it but I make a lot of waste when I butcher a deer, but what I cut always tastes great and the waste comes in handy when coyote hunting