Matrix has always supported "Guest" accounts, where a user can immediately join a room and start talking in there with a generated user_id. They have limitations, but for the "get going straight away" factor it was quite nice.
I personally run https://github.com/turt2live/matrix-media-repo/ which is basically a media server written in Go which has lots of bells and whistles for rentention, storage platforms and scaling.
While Matrix definitely tries to appeal to IRC users, it's absolutely not just a IRC replacement. It's been a conscious decision from Day 1 that history is important given that the default for any messaging system today is to store history. Honestly it feels like a massive improvement over the days of bouncers and logs hosted in the cloud.
That all being said, there are ways to make your rooms drop history if you wish through the config (and I believe it can be configured per room too).
I've never used the feature on Matrix because as you say, it doesn't really apply. Matrix (or really Element here) doesn't require you to use an identity server at all, so I don't really think it invalidates the protocol in a meaningful way.
Yep, it exists as https://p2p.riot.im/! The team is making tons of progress including the new pinecone project. It regularly gets talked about in https://matrix.org/blog/posts if you want to keep up to date.