Cherry pick #66898 to 24.3: [CI Fest] Better processing of broken parts and their projections (fixes rare cases of lost forever)

[robot-clickhouse-ci-1]

Original pull-request #66898

This pull-request is a first step of an automated backporting.
It contains changes similar to calling git cherry-pick locally.
If you intend to continue backporting the changes, then resolve all conflicts if any.
Otherwise, if you do not want to backport them, then just close this pull-request.

The check results does not matter at this step - you can safely ignore them.

Note

This pull-request will be merged automatically. Please, do not merge it manually (but if you accidentally did, nothing bad will happen).

Troubleshooting

If the PR was manually reopened after being closed

If this PR is stuck (i.e. not automatically merged after one day), check #66898 for pr-backports-created label and delete it.

Manually merging will do nothing. The pr-backports-created label prevents the original PR #66898 from being processed.

If the conflicts were resolved in a wrong way

If this cherry-pick PR is completely screwed by a wrong conflicts resolution, and you want to recreate it:

• delete the pr-cherrypick label from the PR
• delete this branch from the repository

You also need to check the original PR #66898 for pr-backports-created, and delete if it's presented there

[robot-ch-test-poll4]

Dear @alesapin, @tavplubix, the PR is not updated for 1d25h0m8s. Please, either resolve the conflicts, or close it to finish the backport process of #66898

[robot-ch-test-poll4]

Dear @alesapin, @tavplubix, the PR is not updated for 1d24h3m26s. Please, either resolve the conflicts, or close it to finish the backport process of #66898

[robot-ch-test-poll]

Dear @alesapin, @tavplubix, the PR is not updated for 1d24h55m48s. Please, either resolve the conflicts, or close it to finish the backport process of #66898

[robot-ch-test-poll2]

Dear @alesapin, @tavplubix, the PR is not updated for 1d24h0m34s. Please, either resolve the conflicts, or close it to finish the backport process of #66898

[robot-ch-test-poll]

Dear @alesapin, @tavplubix, the PR is not updated for 1d24h7m0s. Please, either resolve the conflicts, or close it to finish the backport process of #66898

[robot-ch-test-poll1]

Dear @alesapin, @tavplubix, the PR is not updated for 1d25h1m46s. Please, either resolve the conflicts, or close it to finish the backport process of #66898