Every Cork needs a good Fin(n)ish! 
They all sound Sami (samey?) to me though. 
Seriously, yes, tidy it up or replace it. I use those plastic wine bottle “corks” these days - you can cut them to tube size by warming the male tuning slide of the flute (very carefully) in a gas flame and then pushing it longitudinally down the “cork” (may take a few goes of heating and pushing but much easier than sanding down a real cork! [Doesn’t squeal 
] ), cut off at a suitable length flush across the end of the tube, glaze the end with a hot knife in situ, push it back out with a dowel (may be quite stiff), grease it and pop it in the top end (attached to adjuster/crown if any - just melt a hole in it with a skewer and screw/glue it onto the adjuster prong). I sometimes face stoppers with filed down coins (Euro 10 cents are a good “blank”), but I honestly cannot hear any difference in tone quality. As Cork says, correct positioning is important for the intonation of the octaves.
Footnote: before anyone panics, it doesn’t take that much heat to melt your way through a plastic “cork”. I’ve done the procedure described above quite a few times (more than 12) with no ill effects to the flute head - no cracking, no melting out of tube fixing, etc. You don’t need anything near red hot, and you only need to heat the very edge (end 5-10mm) of the tuning slide/liner tube. You can hold it safely by the wooden head - no burnt hands! Of course, if Nate does it before gluing his liner tube back in, it would be even safer for the flute, though he’d need to insulate his hands on the tube.
Another thought. I recently made a replacement crown & adjuster for a Metzler I’m working on which had lost its original. I hard soldered a piece of threaded brass studding rod (M5 size) onto the centre of a coin, which I then filed down to a snug fit in the liner tube and a smooth outer face (put rod + coin in pillar drill and held files up to it as an improvised lathe). I made a cork as described but melted a hole right through the centre with a heated skewer, then put it over the rod and used a nut and washer to hold it in place. I had a wooden crown turned up by an acquaintance who has a lathe and the necessary skills, with a recess on the inner side into which I fixed another nut with epoxy putty (Milliput) and I filed off the thread on the part of the rod projecting beyond the crown to look tidy. Not quite like the original would have been, but a good substitute and probably more durable.