It was bugged up until 6.21; if the weather shift happened while you were on the island, there was a chance the critter would not appear. Sometimes it would, sometimes it wouldn't. 6.21 fixes that; if you're on the island and the weather shifts at the appropriate time, the animal will spawn.
That said, Island Sanctuary does not actually persist the catch state... or at least, it didn't last time I checked (which, to be fair, was pre-6.21). In simpler terms, this means that if you enter the Island Sanctuary and the weather/time are appropriate, the critter will be there, even if you just failed to catch it and it ran off.
So if you fail to catch the alligator and it runs off but there's still like 10 minutes left in the weather window, you can go back to Moraby and then return to your island, and the alligator will be there again. Which is a detail that likely comes too late to be useful to OP, alas, but if someone else misses the alligator for the same reason -- e.g., you couldn't release an animal -- you could quickly go free up a slot, leave the island, and return to make another attempt before the window ended.
While I disagree with Deveryn, I will point out the statement made was that the game tells you how to end the caretaker service -- which is true. It does, your screenshots even show that. I just disagree with Deveryn that "how to end the caretaker service" is the relevant issue here, because so far as I can tell no one is confused on how you discontinue caring for an animal.
The issue is that it doesn't tell you that the capture process changes if animals are under care; that an animal being under care cannot be released, meaning that if you have a full pasture and catch a rare spawn you have no way to actually accept it. (As opposed to not having animals under care, in which case you can pick a release from the window.) Moreover, the first time it tries to communicate this is when you get the capture interface pulled up and then it says you can't actually release anything from the list it gives you.
This is actually a recurring issue in the game, where UI will blithely offer you a choice which the system can reasonably know is not viable at a point where it is either inconvenient (or with this particular issue, actually impossible) to remedy the situation.
An untradable minion you already have, or loot after you've already won something for the week in a limited-loot environment? The loot window still forces you to roll 'Pass' on it, despite the game knowing there's literally no other option. You click on 'Greed' and it throws up an error message, and you then have to click Pass. Why does it even allow me to roll on things where the only valid option is Pass?
Free Company submersibles also suffer from a form of this. A submersible will come back, and I can click 'Redeploy' to send it out again with the same route... and then it will tell me the submersible needs repair, forcing me to click Finalize instead, then go to "Repair submersible components", then go back to "View previous voyage log" and now click "Redeploy". The system knows the submersible needs repair; it immediately rejects the Redeploy because of that. It ought to just grey out the button (or even better, replace it with 'Repair') when the submersible can't be sent back out.
I understand why it does this -- the loot window is kept as simple as possible, the submersible log window is kept as simple as possible, etc. You can then offload error handling into something else which can be used from multiple places. I just disagree that it's the right approach; you could make the error handling a yes/no check ("Can you roll on this?") utilized by the UI, rather than a function you call ("Roll on this") which returns an error ("You already have one.").
This isn't quite the same scenario, but it's certainly similar.