AWS separates the anycast LB functionality into a separate service called AWS Global Accelerator. You do get individual anycast IP addresses with that service.
Ah, interesting; it's been a while since I played with AWS, and that service wasn't there back then. I'm guessing that allocating a new AWS Global Accelerator address takes a while?
I've only done it once (the way they have it architected, it's a "set and forget" sort of thing, your LB changes don't touch the Global Accelerator) but I do seem to recall that it took awhile to create the resource. Maybe 5-10 minutes?