Except the Skyrocket wasn't ready for production until 1942, at which point it was competing with Hellcats and Corsairs. That's why Grumman moved the concept on to the Tigercat.
First flight was one month earlier than the F4U, while the F6F was a rush job which only made its first flight in 1942. So, you're right that it probably would have been a stable-mate of the Corsair and the Hellcat had it gone into production.
I think that the only basic problem was oil cooling, though, which was fixed easily enough. More extensive redesign was Grumman's attempt to keep the Navy's interest and/or get the Army Air Force to buy it. Optimistically, it could have been ready for service by early 1942.
Given its not great service ceiling, it could have had real trouble in its intended role as bomber interceptor, and who knows if early war U.S. pilots would have figured out how to use it properly. That, plus the fact that groundcrew would have to maintain twice the number of engines, and it would have taken up more space in aircraft carrier hangers probably sealed its fate.