Heh. Well, I figured I'd bump this and post what happened.
First, thankfully the landlord was nice about it, and fixed it. But when the workman came, I had to leave midway through; and I didn't notice until days later that the man had made hardly any attempt to remove the old caulk first. And, apparently, hadn't done much about the moisture and calcium deposits located under the old caulk, and now under the new.
Anybody who knows how this works, can predict that it wasn't even a week later when the new caulk started separating from the bathtub. I bet the man didn't even fill up the tub before he started; it wasn't full when he left, and the caulk couldn't have been set by then, since that takes a couple of days.
But before I'd left, the workman had also mentioned that the landlord wasn't happy about paying him for these renovations.
So, when I found out just how badly this had been botched, instead of bothering the landlord about it and having to go slightly nuts through another conversation about repairs, I went and bought myself some caulk, went at the old gunk with a scraper, dried it out, and put in the new. Total cost... maybe $5-$10, somewhere around there. I hope I did it right. I've used caulk before, I know how it behaves, and it looks good... but it hasn't set yet and I really hope there wasn't any extra moisture trapped in there or anything.
And I'm petrified that the landlord will drop by tomorrow, and see the bathtub filled with water and new caulk just applied to it. The workman seemed like a nice guy, just probably some random guy they hired who wasn't really a professional... Well, I'm not a professional either, but I know to cut out the old stuff before you put in the new...
Yeah. So there's that. I'll just have to hope my repairs hold and they never figure out the difference.
And somehow get through another two or three days without a shower. Sponge-bathing at the kitchen sink might not be as annoying as the first prickly spray from a shower head, but it really doesn't get you nearly as clean.