Based off of what we see of the red lion so far, it doesn't need a pilot in order to find keith. Based off of the traits that the Red Lion is suppose to have, I don't think it would need someone who doesn't fit it's traits in order to save Keith, unless Keith is in serious dire help, in which case Lance having the time of his life doesn't really seem necessary. So Lance in red and having the time of his life doesn't make sense, because Lance doesn't have the traits that allows him to pilot it
I would contest this, mostly because, simply enough- the Lions have to have some benefit from having a seated pilot, otherwise we wouldn’t have a show. We also see in practice that it is very rare, and often evidence of a dire situation, that a Lion attacks or fights on their own, and we’ve never seen a Lion in single combat- the closest we’ve come is Black one-shotting Zarkon out of the astral plane. Red’s assault on the base wasn’t against another enemy, just a stationary target.
Yes, Red hasn’t taken a pilot with her before to go find Keith, but Keith has never been under fire before, he’s just been lost or floating out in space away from enemies. Rescuing Keith in the middle of a firefight would involve Red fighting on her own, and, when it comes to fighting, even with fights where the Lions easily outmatch their foes?
Black still needed Keith to work with her to rescue Shiro from those saber-toothed creatures. Even though Black could basically step on them and hit them with her tail to do so.
That much is given to us by canon- the Lions need a pilot, and sometimes, in desperate straits, any pilot will do even if it cannot work optimally. They would much rather wait or hold out for their designated paladin, if they can choose, but sometimes they can’t. Compare Red ignoring Coran to go get Keith, to Black accepting Keith because Shiro was away from her and about to get mauled by lizards- she couldn’t wait for him to run her way, and, seemingly, sustaining damage from the wormhole and Zarkon’s tampering, she couldn’t go get him on her own.