The way I see it, it isn't really "killing", as in, getting it to not exist at all anymore. That old "you" will always be there, just like a bunch of other possible "yous". The "killing off" part means getting to a point where going back to that old "you" becomes as difficult as it was for you to get to the current "you" from your old "you".
And I totally feel you on the part about thinking too hard about it. This outlook I have is definitely not comprehensive and just provides satisfaction at a cursory glance when my mind wanders there. Looking at it under a microscopic view, there are individual parts that can be deconstructed and found to be not super accurate. But overall, it seems to provide me with a good enough framework to look at my life through.
I definitely prefer the new me. There is a continual spectrum of mes from being a young kid to an adult (ship of theseus style[1]) but this is different (and I think Elton John's situation was similar). I can clearly see a me I like and an alternative me. In theory, every day of my life I make a choice to give that day to the me I like - but gods when I think about it it can get heart wrenching. Digging in, I'm basically denying that alternative me a chance to be alive... And I've never killed that other me, it's sitting there and I might be that me tomorrow, it's just that every day so far I've chosen that today won't be the day it is me... Similarly Elton John could always return to being Reginald (or however he identifies his older identity) he just doesn't.
As someone who waxes philosophically this is an interesting consideration to dive into but thankfully most days I skip right past it and don't perseverate. It's pleasant to share though, since I don't like vocalizing it with most folks.
1. https://en.wikipedia.org/wiki/Ship_of_Theseus