I hope the current D208 are something else than the Sportmax D208s I had on the CB400 back in 2013! Most slippery rubber I have ever ridden on, by a big margin. They were original equipment on the DRZ400 motard allso, and may have worked on 100F sunny days, be we do not get those here. Ever.
Sport touring tires work quite well when cold, and also warm up much quicker in my experience, compared to sport rubber. Even touring type bias ply will out-perform sport rubber under some conditions. I'll bet that on a day with 20-30F, something like a set of Metzeler ME77s will offer a LOT more grip than any modern sport radial made. At least with street tire pressure.
In short, there is nothing wrong with sport rubber, but they are made for a purpose. Riding on public roads at respectable (as in legal) speeds in chilly temps are not such a purpose IMHO. In fact, even on a hot day, at a sensible pace for public roads, I doubt any pure sport tire will offer the grip of a sport touring model.