It's done now except for figuring out how to correct the previously unknown structural issues described in #11 (essentially a void in the concrete and a crack, both of which are covered over by tile and only accessible via a 1/4" hole, post #17 has the current plan - but that could be revised if someone has a better way).
To fill the void under the step it looks like what I want is anchor cement. Not sure I'll do anything with the other issue (assumed to be a crack) as the screw pulled tight, but it seems like anchoring epoxy would be best.
Filling the void is probably about a cup (hard to say, my "probe" was a 5" twist-tie and I envision it as the corner
broken off a concrete step ; -), how I'm going to do that through a 1/4" hole is problematic. The tile is 1/4"
thick too, and naturally, it's on the side against the wall so it's the hardest spot to work in. Also worried about it
"expanding", don't want to break the tile.
The anchor cement looks loose enough that I can pour it into the back of a caulking tube for "injection", but
ideally I'd want a 3/16" tip so the hole can "burp". Not crazy about the quick setup time (suppose I could
experiment with a little retarder and see if flow-control would make it runnier).
The default tips (see right image) look too big. A cooking syringe might be ideal, but not sure I'd have the hand strength....depends on how viscous the concrete is.
Possibly a short 3/16" OD tube connected to the end of the caulking tip and pushed in the hole?
The filler will have to bond with the concrete behind the void, and that won't be easy to do. The best way to get a solid anchor is to use longer bolts. Now if you want to put weight on the tile, it might crack, in that case filling the void completely would be a good idea.