task_job_mgr: make try/except more strict

[oliver-sanders]

• Address a code TODO to reduce the scope of a try/except to the individual expressions it was intended to cover
• The overzealous error catching had hidden a genuine error causing it to be missed for some time, see polling/set: submitted task cannot be set to succeeded #6314

Check List

• I have read CONTRIBUTING.md and added my name as a Code Contributor.
• Contains logically grouped changes (else tidy your branch by rebase).
• Does not contain off-topic changes (use other PRs for other changes).
• Applied any dependency changes to both setup.cfg (and conda-environment.yml if present).
• Tests are included (or explain why tests are not needed).
• Changelog entry included if this is a change that can affect users
• Cylc-Doc pull request opened if required at cylc/cylc-doc/pull/XXXX.
• If this is a bug fix, PR should be raised against the relevant ?.?.x branch.