[v26.1.x] sr/context: implement /contexts/{context}/... prefixed handlers

[vbotbuildovich]

Backport #30189 to branch v26.1.x. This issue was created as a placeholder because the original PR's commit(s) could not be automatically cherry-picked.
Here are the commands to execute:

git checkout v26.1.x
git checkout -b manual-backport-30189-v26.1.x-299
git cherry-pick -x cceffc0499 c1506c29a3 5551d4325d 913cedb4c8 0acf252efd a93647dd4a c44d2c3913 8e2b44b2c7 7c2d3d8bfa 18720e284c cb5aff6ec9 955d7bec2b e314131f65 bc778c4455

git push origin manual-backport-30189-v26.1.x-299
gh pr create \
  --title "[v26.1.x] sr/context: implement /contexts/{context}/... prefixed handlers" \
  --base "v26.1.x" \
  --label "kind/backport" \
  --head "manual-backport-30189-v26.1.x-299" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v26.1.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/30189 "