After updating Hamachi to the latest version, created games stopped appearing altogether.
So, the question is - what could be causing the crashes? Where should I look for a solution?

I can't say for sure about the crashes - it seems to be a random thing... it depends, of course, on the provider, internet speed, etc.
There are certain patterns regarding the versions.
I've used Hamachi 1.0.1.5, 1.0.2.5, and 1.0.3.0.
In my experience, 1.0.1.5 is the most stable.
The newer the Hamachi version, the better it is protected, but this probably causes problems with "seeing" the game.
As for the game versions, 3.2 vs 3.2 and 3.2 vs 4.0 have a good connection, but 4.0 vs 4.0 sometimes shows some quirks.
And if you can't see a created game in Hamachi, check the IP address of the "creator" in Hamachi and manually search for the game and enter the IP address there.
P.S. I'm currently using Hamachi 1.0.1.5 and Heroes 3 "Platinum" 4.0.