Case creation in Lightning - Send notification email to contact option not sending email when Process updates record
At Derek's suggestion I'll turn my comment into an answer:
The problem sound like the known issue Send Notification Email to Contact on Cases does not send email. If I remember correctly this issue had been in review for quite some time, so requesting an update from support may unfortunately be in order.
There's also this known issue, which sounds similar, and may have the same underlying cause: "Send Email Notification to Contact" failing to trigger email in lightning when flow is attempting to create associated asset.
The last updated date on that case is 2019-06-27 (currently 7 1/2 months) and it's still in review. Depending on your patience, you may want to request an update on that case as well.