Based on what I've been able to find, the first use of "Let's Fly!" (Burnham's catchphrase) was in the season 3 finale, "That Hope is You Pt. 2", first airing on Jan 7, 2021. Capt. Freeman's "Warp Me!" appeared in S1 E9 "Crisis Point", airing October 1, 2020 - technically around 3 months sooner than its DSC companion episode with a similar concept. Both captains were ruminating over what their catchphrases should be in multiple episodes leading up to their final decisions (no idea when the dialog started).Also, wasn’t Michael’s line in season 4?
So, it IS possible that DSC came up with the concept in the writing room sooner and one of the LDS writers heard about it and decided to do one of their own, poking fun at it, and their episode just happened to drop before DSC's did. Then again, it's also possible that LDS was the first and DSC followed suit to try to be a little more trendy.
After Orville hit the air, it has always felt (to me) like DSC has been trying to play catch-up, especially when they realized it's actually okay to have a dramatic show with some slapstick comedy (Linus sneezing?) They saw the head-scratching success of LDS amongst the fandom and decided to continue to draw from the comedy well.
I would love to see someone's severed leg fall out of the Enterprise's ceiling. One of Chapel's "experiments".
