[v23.3.x] tests: wait for the leadership change metric to be updated

[vbotbuildovich]

Backport #20813 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-20813-v23.3.x-997
git cherry-pick -x e62d561baf31098af0675ccbaa567021d897cc38 877276383c8badbaaaec8af759ecb91c7d72b8d4 2c009b1771e91422e42a401dde29e8a0f8ccdc6a

git push origin vbotbuildovich/backport-20813-v23.3.x-997
gh pr create \
  --title "[v23.3.x] tests: wait for the leadership change metric to be updated" \
  --base "v23.3.x" \
  --label "kind/backport" \
  --head "vbotbuildovich/backport-20813-v23.3.x-997" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v23.3.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/20813 "

JIRA Link: [CORE-5099](https://redpandadata.atlassian.net/browse/CORE-5099)