You're right that most of the time we need non-trivial algorithms or visualizations for trees for it to be useful. If I ignored that it would be like wishing for a "general binary editor" because it would be so "powerful" and all data can be represented as binary.
I think, though, that there's enough common ground between the various domain-specific tree editors that some kind of useful intersection exists. Then the domain-specific tree editors would simply be plugins.
Perhaps the "killer app" is an extensible GUI framework with some sensible defaults already implemented (e.g. those mimicking some of the design dimensions explored by the software mentioned in other posts), combined with a scripting language and/or a suite of good libraries/apis for popular languages.
I think, though, that there's enough common ground between the various domain-specific tree editors that some kind of useful intersection exists. Then the domain-specific tree editors would simply be plugins.