[v23.3.x] admin/debug/leader_info: prevent reactor stalls

[vbotbuildovich]

Backport #17473 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-17473-v23.3.x-711
git cherry-pick -x 7879dc20333ee2daa7b50e6a2a2826eb4b9bfe20

git push origin vbotbuildovich/backport-17473-v23.3.x-711
gh pr create \
  --title "[v23.3.x] admin/debug/leader_info: prevent reactor stalls" \
  --base "v23.3.x" \
  --label "kind/backport" \
  --head "vbotbuildovich/backport-17473-v23.3.x-711" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v23.3.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/17473 "

JIRA Link: [CORE-1997](https://redpandadata.atlassian.net/browse/CORE-1997)