I would like to know why the Gorn was there too, but the Gorn didn’t notice Spock boarding and so there’s no reason to believe Enterprise would have notice a Gorn boarding. It’s also possible it was beamed over when the Cayuga was destroyed to find something or to access the ships systems.
Janeway also attempts to give CPR to a dead crewman in Voy: Scientific Method
+1 for the complaint on the ‘do the thing’ comment. I feel like it’s because engage/make it so have become somewhat of a meme from Picard in TNG and they’re looking to replicate that.
But the idea every captain has their thing is one seemingly from the new Treks. In the past series, engage is used frequently by Kirk, Janeway and Sisko. (Kirk also frequently uses warp speed Mr sulu, ahead warp factor 1, take her out, first star to the left and straight on till morning etc.) They’ve been really pushing ‘let’s fly’ from Discovery, but it’s a terrible line and trying to force it just makes it cringy. I wish they’d used that time to expand the stealing the Enterprise plot a little more.
Cayuga is a connie according to memory alpha - confirmed in some background graphics: https://twitter.com/timothypeel1/status/1690018124492349440/photo/1