[v23.1.x] dt/many_clients: Split out test_many_clients

[vbotbuildovich]

Backport #11385 to branch v23.1.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 v23.1.x
git checkout -b vbotbuildovich/backport-11385-v23.1.x-429
git cherry-pick -x ba3326ddbb26c06d169b7d4bb3f71d1783e2c4cb

git push origin vbotbuildovich/backport-11385-v23.1.x-429
gh pr create \
  --title "[v23.1.x] dt/many_clients: Split out test_many_clients" \
  --base "v23.1.x" \
  --label "kind/backport" \
  --head "vbotbuildovich/backport-11385-v23.1.x-429" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --reviewer "" \
  --milestone "v23.1.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/11385 "