The content type that is being displayed on View Properties should inherit "Document Id" field. In case, for some reason, if this is missing, the "Document Id" field won't be visible.
We can add this back using Server Side API. Goto the ContentType (Hub CT or list CT doesn't matter) add all the "Document Id" related fields using FieldLinks property. The changes made to the CT here are scoped to that list only.
The assumption is the "Document ID" field is already available in the list but just not showing up in the document properties.
$ct = $list.ContentTypes["your content type name"]
$ct.ReadOnly = $false
$spDocIdField = $list.Fields["Document ID"] #Ideally adding this field is enough to see the document id in view properites
$spDocIdField = $list.Fields["Document ID Value"]
$spDocIdField = $list.Fields["Persist ID"]
$ct.ReadOnly = $true
In other words, this is kind of replicating a field to "Add to All Content Types" functionality when introducing a new field in the list.
The OOB way to fix this issue is republish the content types and re-run the CT Hub sync and Doc Id timer jobs.