A widget-like component, such as the voting feature, would be useful. Perhaps you could start using public voting to show which people are in need of lessons and which people offer lessons.
We aim to use voting in this way. At least, people can change their vote without causing further noise in a channel.
It would be beneficial if the voting plugin had an option to not sort the results for this kind of use case.
Different thought: You could experiment with the easily extendable oneboxing feature to include external tooling into the site.
Brainstorming, how a tooling may look like:
Available time for users seems to be independent of topics, lessons, or machines. Users could be able to announce time slots in their user profiles. Perhaps they can paint in a week overview to show their repetitive usual available time slots and add dedicated time slots to their own calendar. These announcements could have different statuses, such as “will be there” or “might be there.”
This general information could be used together with information about skills (who is able to provide which skill, who is in need of which skill) and maybe also available time slots for machines in a second stage.