I do not think the Internet ever was like that but maybe you were using a service like AOL or Compuserve which assumed you were email another member of the same service unless you explicitly put something in the address like “internet:sailor@yummy.cat”
It sounds like you’re talking about a bang path address, a very old style of email address. The address worked by specifying a route to the recipient in the form of a series of machine names separated by bangs (“bang” is Hackish for “exclamation point”), with the final term being the recipient’s ID on the final machine in the route.
This was back in the days before every company had internet access everywhere. If I wanted to send an email to someone outside the company, I had to use a bang path which specified the hops to get to one of the servers which had outside access. Made for some really long addresses and was a major PITA.
Back in the day I had an account on one of the backbone machines (decvax) which made it very easy for people to send me mail, since most everyone could get to a backbone. Then I could reply to their mail without ever learning their complicated address.
The example given in the OP is a mix of Arpanet and UUCP addressing. Pure UUCP would be like:
host1!host2!host3!username
So from “host0” (my site) it would go to host1, then host2, then host3 where username lived.
IIRC, the form in the OP developed later to access someone on the Arpanet. It would go to host1, to host2 which was on Arpanet, which then sent it via Arpanet to host3. So host2 was acting as a UUCP to Arpanet gateway. (And you could go the other way as well. It could get complicated, and slow.)