ISSUE
As of the Spring '23 release, the Case Owner field was exposed on the Email-to-Case routing address settings page, specifically for the source type of Email-to-Case; previously, the field was only exposed for the source type Outlook, which is used for the Salesforce for Outlook product Create A Case Button.
If the organization has active case assignment rules with email notifications configured, in addition to a populated Case Owner value on the Email-to-Case routing address, the case assignment rules will be ignored.
As a result, the case will be assigned to the configured case owner via the Routing Address settings; however, no new case notification email will be sent out.
Control Email-to-Case email notifications:
000381861

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