Why is it when I join a new company, the people I depend on to train me on complex systems suddenly turn in their two-week notices and quit? This ALWAYS sets me up for failure because people expect me to instantly BE that person who quit. That’s impossible and unreasonable.

That sounds fun, ..

Thankfully I never had that happen to me before, though now I'm scared ..

Stop talking about your sicilian friends and that you would like to inherit your mentor's position.

Because the company likely hired you to replace another person that quit a month earlier.

If ypu aren't fortunate enough to be being taught by someone who wants you to know what they know, the onus is on you to force that process. You know what you have understood and become capable of dealing with. Ask for repeated and further clarificarions of everything you have not. Push to be informed of anything not in your list of things you have comprehended. If not, as they say, it's your funeral...

It's likely that they gave 2 weeks notice prior to your hiring, but your manager/boss convinced them to stick around until they hire a replacement and then train you for two weeks - this is quite common because they usually double the pay for the person who is quitting just so they can at least somewhat train the next person.

But now and then you get a good experience like mine: When I was interviewing for a lead dev position the guy that I was replacing was there and told me, I'm starting my own company, but I'll be here for another 3 months to help you with anything you need and even after that you can still contact me if you need any help.

Total honesty on their part, and they kept their word this guy showed me as much as possible and he still answers to every question I have and even randomly sends recommendations from time to time.