More deprecations

[MikeMcQuaid]

More deprecations for the next major/minor release.

• Have you followed our Contributing guidelines?
• Have you checked for other open Pull Requests for the same change?
• Have you explained what your changes do? Performance claims (e.g. "this is faster") must include brew benchmark results.
• Have you explained why you'd like these changes included, not just what they do?
• For bug fixes, have you given step-by-step brew commands to reproduce the bug?
• Have you written new tests (excluding integration tests)? Here's an example.
• Have you successfully run brew lgtm (style, typechecking and tests) locally?

---

• I did not use AI/LLM to create this PR, or I disclosed the tool/model below and reviewed its output; I did not attribute commits to AI and will answer maintainer questions and review comments myself without AI/LLM.

Codex with GPT 5.6-sol at xhigh effort, with local review and testing.

---

[samford]

Now that I've worked through all the related fixes, I opened a draft PR for the livecheck user agent fallback deprecation (#23834). I had been sitting on the deprecation changes while I worked through fixes, so apologies for the delay. I saw this PR after the fact but it may be better to have the deprecation as a separate PR anyway, as there are some fixes that still need to be merged before the deprecation can be merged or it will lead to failing checks (and I don't want to hold up your PR).

That said, the approach I took differs in a notable way, where the deprecation is only triggered if the check succeeds with the fallback user agent. The idea is that triggering the deprecation for a check that fails regardless of user agent wouldn't be useful/applicable (i.e., it would mislead users). [I also seem to have taken a different approach to the tests, for better or worse.]

Let me know what you think but please don't merge either user agent fallback deprecation until the fixes linked in my PR are all merged.