[v23.3.x] [CORE-60] Schema Registry: Support /mode

[vbotbuildovich]

Backport #17952 to branch v23.3.x. Note that this issue was created as a placeholder, since the original PR's commit(s) could not be automatically cherry-picked.
Here are the commands to execute:

git checkout v23.3.x
git checkout -b vbotbuildovich/backport-17952-v23.3.x-452
git cherry-pick -x c99992877894ada0f7a69878b4c42ab8ec4f4519 5fb72fcf3d840e4290417d62ed0bfb9db7c508f9 452556b74f8d3afd29198308c9c60414a87d1df4 88e1993bb1c6c0677668cbc0134de489b65d1eed ba05b97ce676bf276cd25b0eee16bbc03b2b9360 390fc71805c0b131f5d3154af4b508e730303827 7aa34ffccee787e6d725b621ebe7fdd77befe1f5 71c5fef6fd374ddba669fde6edad32e1318cd607 6ce966f23e086138f6f18368f1ef17799fb374fa 2776733646c541d335085e36485c993a797a0f0b 282854b2223cd8b423318dac62367e5520b41ba9 8b0915f8e22d9768c5f0458c80048ee4f847af3f b55d09ed7a031345182efe5f45272d118cea0fe8 ccdb7d6f060a66417f6cae7c5828d33e60a67f78 6f2f894891c146fc56d7f57bce868b7930db13ad dc8c2e429a1c23db2d87eaa834ff67d6faebb468 dd6c8312fa26a237af40e5d447ea35d2367474fa 19e51c9b6695f9a81fb777d297e511ec3095ced6

git push origin vbotbuildovich/backport-17952-v23.3.x-452
gh pr create \
  --title "[v23.3.x] [CORE-60] Schema Registry: Support `/mode`" \
  --base "v23.3.x" \
  --label "kind/backport" \
  --head "vbotbuildovich/backport-17952-v23.3.x-452" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v23.3.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/17952 "

JIRA Link: [CORE-3150](https://redpandadata.atlassian.net/browse/CORE-3150)

[michael-redpanda]

Fixed by #18742