From a technology perspective, it could be used for either instant messaging or email-like conversations. The common use cases right now place it more in line with realtime messaging like Slack. We just launched some documentation about Matrix[1] on the Ruma site that explains pretty thoroughly why someone would want to use Matrix, why someone would want to run their own homeserver, and quite literally "Why Matrix Matters"[2]. In short, the value is being able to (eventually, once the software matures) use a single client application to chat with all your contacts across all the various chat networks, and to maintain ownership and control of your own data.
The "Why" page comes across to me as a bit of an activist pitch. You can be right (and you probably are) about limited consumer choice, and the dangers of corporate consolidation, and the lack of privacy, but none of those things constitute a selling point to the average consumer. If they did, then the market would at least be in the process of solving these problems for you, and I don't see any sign of that at all.
These things do matter to businesses though. I've seen multiple companies adopting Slack express discomfort over the idea that the communications are hosted and archived by Slack. Email is broken and horrible, but businesses still use it for a lot of the same reasons they might choose Matrix. Slack fixes the horrible, but lacks (and its business model is likely incompatible with) a federated protocol. There could be an opportunity there.
That wouldn't be one company though, that'd be hundreds of different companies all providing their own clients and hosting solutions.
All good points. It will probably be helpful to have a bit more context on the goals of Ruma as a project. It is a passion project, and has no intention of ever having direct financial backing or a business model. The "Why" page sounding like activism is somewhat intentional—it explains the motivations for the project, but it is not specifically intended to "sell" it to a person who is not already sympathetic to the problems it tries to address.
Perhaps a good comparison is the public reaction to Snowden's leaks about the NSA. They haven't been super big news outside of tech and certain political circles. The average person values convenience and a good user experience over any more philosophical or political beliefs related to privacy and security. The idea of a centralized service or company controlling data just doesn't matter to a lot of people. You're absolutely right that the business motivation is not quite there, because there isn't a strong public demand for it.
Matrix itself does have some financial incentives—specifically, its development is funded in part by a company called OpenMarket, who are also the developers of Vector, a web-based client for Matrix that will at some point have a commercial product offering.
Edit: In other words, Vector is planning to do exactly what you describe: provide a product that competes with Slack, but building it on a protocol where someone could create a competitor using the same protocol, allowing them to be interoperable. Ruma does not have a financial stake in the game, and I've chosen to build on Matrix because it is well aligned with my values.
[1] https://www.ruma.io/docs/matrix/ [2] https://www.ruma.io/docs/matrix/why/