feat(mise): automagically support all tools through Mise Registry

[jamietanna]

Changes

As noted in #42250, we should try and auto-support tools from Mise where
possible.

Similar to other areas of the codebase, we should use the "static data"
pattern, and periodically sync the mise registry --json output with
lib/data/mise-registry.json, and then auto-detect the tools that we
support through the known backends the tools support.

This takes lower precedence than existing in-code tool definitions, so
we can still provide an "override" if we can support lookups or version
extraction more appropriately.

Because we now have many more tools we (mostly) support out-of-the-box,
we need to provide an improved documentation page, and:

• only link to tools we have an explicit tools URL for
• note where a backend may not be supported

To automate the updating of this data, we:

• make sure we have mise in GitHub Actions, before we run
• don't pin the mise version to make sure that we're always seeing the
  latest supported tools
• make sure the commits generated are now feat(...), as they introduce
  functionality changes (in the case new tool(s) are supported)

Note

This improves support from ~170 tool short names to just over 1000.

Context

Please select one of the below:

• This closes an existing Issue: Closes feat(mise): auto-generate supported backends #42250
• This doesn't close an Issue, but I accept the risk that this PR may be closed if maintainers disagree with its opening or implementation

AI assistance disclosure

Did you use AI tools to create any part of this pull request?

Please select one option and, if yes, briefly describe how AI was used (e.g., code, tests, docs) and which tool(s) you used.

• No — I did not use AI for this contribution.
• Yes — minimal assistance (e.g., IDE autocomplete, small code completions, grammar fixes).
• Yes — substantive assistance (AI generated non‑trivial portions of code, tests, or documentation).
• Yes — other (please describe): Claude Sonnet 4.6

Documentation (please check one with an [x])

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• No documentation update is required

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I have verified these changes via:

• Code inspection only, or
• Newly added/modified unit tests, or
• No unit tests but ran on a real repository, or
• Both unit tests + ran on a real repository

The public repository: JamieTanna-Mend-testing/mise#2 wasn't supported before