[v24.2.x] Fix stepping down on timeout

[vbotbuildovich]

Backport #24590 to branch v24.2.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 v24.2.x
git checkout -b manual-backport-24590-v24.2.x-643
git cherry-pick -x 6a1e34bead 95a29dba65 5f69d9b733 7d33bb5659 f04995a751 8b57b42101 67e7c6ea21

git push origin manual-backport-24590-v24.2.x-643
gh pr create \
  --title "[v24.2.x] Fix stepping down on timeout" \
  --base "v24.2.x" \
  --label "kind/backport" \
  --head "manual-backport-24590-v24.2.x-643" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v24.2.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/24590 "

JIRA Link: [CORE-8697](https://redpandadata.atlassian.net/browse/CORE-8697)