[v23.3.x] xform/service: Remove exception handling from _log_manager->start

[vbotbuildovich]

Backport #16571 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-16571-v23.3.x-258
git cherry-pick -x 84dac04e55f35f0c8f615be1823405f32c2bd804

git push origin vbotbuildovich/backport-16571-v23.3.x-258
gh pr create \
  --title "[v23.3.x] xform/service: Remove exception handling from _log_manager->start" \
  --base "v23.3.x" \
  --label "kind/backport" \
  --head "vbotbuildovich/backport-16571-v23.3.x-258" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v23.3.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/16571 "

[oleiman]

Pick onto #16570

[oleiman]

closed by #16663