That was a big drawback (which was later remedied). An odd oversight, considering earlier dogfights with MiGs during the Korean War.
There was a period of time - in the late ‘50s and early ‘60s, I believe - during which accepted Air Force doctrine and policy was that there would be no more dogfights. That dogfights couldn’t happen anymore because closing speeds and angular rates were so high that human reaction times wouldn’t be able to deal with a dogfight.
Air Force thinking was - for a time - that everything would depend on air-to-air missiles and other stand-off capabilities.
Perhaps the lack of guns on the Phantom was a result of this policy.