[v23.2.x] [CORE-96]: admin/server fix set_log_level handling of overlapping expirations

[vbotbuildovich]

Backport #18397 to branch v23.2.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 v23.2.x
git checkout -b vbotbuildovich/backport-18397-v23.2.x-221
git cherry-pick -x bf6bc91ae98162e9504cb33fbffde62f5045211c 0717cd1a0c9f352ad8a66d6cdf10ae58f36147a1 6593dbec194446702799c3223caeb688000019cf d781fb13e6051bf3b01821b9faac4687fa69f1c6 84748a8dcb4931258c5b9ef0cb3a32b9f2b9202b

git push origin vbotbuildovich/backport-18397-v23.2.x-221
gh pr create \
  --title "[v23.2.x] [CORE-96]: admin/server fix set_log_level handling of overlapping expirations" \
  --base "v23.2.x" \
  --label "kind/backport" \
  --head "vbotbuildovich/backport-18397-v23.2.x-221" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v23.2.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/18397 "

JIRA Link: [CORE-2935](https://redpandadata.atlassian.net/browse/CORE-2935)

[andijcr]

closing this as the bug is not in v23.2