From the looks of the broken screws, they have actually been broken for some time - there's only a tiny bit of bright brass colored metal, the rest is well-oxidized. As you know we're not using the brass screws for many years now. I'll be happy to send you a set of the stainless screws (email me your shipping address, I already have them packed up for you).
But of course, you'll need to get those broken brass screws out. The reverse-thread screw is an excellent suggestion, but if it doesn't work, you'll need to drill out the screw, leaving a larger hole. It really doesn't matter since you'll have to fill the hole either way.
Once you fill the hole with a dowel and titebond wood glue, then drill a pilot hole for the stainless steel screws and replace the tailpiece.
I too am relieved that there wasn't more catastrophic damage. I'm also a little curious about the silicone, nobody here remembers using it in that location before. Learn some bit of Alembic trivia every day!