"If this shirt delivers on its promise without any showstopper-sized downside, I'll be in the market."
They already exist, give a wee google. 30-45 euro a pop, labfresh and underarmour are pretty good.
Personally I go for thick cotton ones, wear them for half a day, hang dry and biff them in the freezer. If I'm outside and sweaty then merino, but they are not a cheap option. Plus if keeping warm is an issue, then smelling of sweat isn't.
Then again, these days I do housekeeping, where everyone is sweating by 1300, and bouldering, where everyone is sweaty.