Narratively, Goku using the Kaio-Ken could have overshadowed Super Saiyan 4, so it makes sense why he doesn't implement it. From Goku's perspective, however, it would have given him a clear ...

Goku's Super Saiyan Blue with Kaioken combines two techniques, greatly increasing his power but straining his energy and stamina levels. Vegeta's Super Saiyan Blue Evolved is a new transformation ...

Perhaps Goku felt that using Kaioken while in his Super Saiyan Blue form simply wouldn’t be able to overpower Broly fast enough for him to not overcome it.

Kaioken, of course, first appeared in the Saiyan Saga of the Dragon Ball Z franchise, with Goku learning it on King Kai’s planet following his death to his brother Raditz.