I'll give this a shot, what you're saying though sounds like you do it with a Tune-O-Matic, I'm using a Strat with a non-hardtail.
Actually, on a Floyd Rose. I'm a strat guy too though, and while I haven't had to do it on mine, I see no reason why it shouldn't work.
Moving the saddle shouldn't make a difference.
One question that I don't believe has been asked yet - where EXACTLY is the string breaking? Where the string crosses the saddle, inside the trem, near the ball end? "The core snapping, the winding coming undone" sort of makes me thing that it's the later, and that it's happening right at the very end, inside the trem block.
If so, the saddle isn't your problem (though, polishing it certainly didn't hurt things). One of two things could be happening -
1.) there's something funky with the trem block, or you're not seating the ball end correctly into the cavity. Take a look in the back with a flashlight and see if anything looks funny on that string.
2.) You could have just gotten a bad batch of strings, too.
I was super-excited and all Fender fanboi when I bought my Strat, a bit over a decade ago at the ripe old age of 15. I forgot what strings I was using at the time, but I immediately switched to Fender, because, well, Fender suggested it and I
owned a Fender so it only seemed right.
I quickly realized Fender strings pretty much blew. They were really bright and present at first, but almost immediately died and sounded like rubber bands. I ended up experimenting a bit and switching to Ernie Balls because they sounded more balanced on that guitar, and held up a little better, but these days I swear by Elixir Nanowebs. They're pricy, so I wouldn't try them until you get your breakage issues sorted out, but certainly trying a new brand seems like a good idea right about now.
BTW, is this a vintage style "bent metal" saddle bridge, or a more modern "cast saddle" type?
bent metal:
cast metal:
EDIT - and what a shitty intonation job.
