Same problem here. Wish I’d known before I uninstalled all my apps, reinstalled the Steam version, tried Crossover Preview, and wasted 2 hours on all that. Turns out, it’s not even a Crossover issue—it’s on Blizzard/Overwatch’s end. This bug is all over Windows too: Battle.Net saying "Playing Now" when the game isn't opening (plenty more examples if you search around).
There’s no real fix for it—just gotta wait for a new patch. Some people say you can leave it running, and it’ll open after about 10 minutes, but no luck for me. Pretty weird stuff.