Email button is not showing since you are using global email action in the Quick Action.
If you can closely look into the properties, you will see like this.
- Remove that action from page layout
- Go to Support Settings and choose
Enable Case Feed Actions and Feed Items:
When you select this option, you will go the feed based page layout and will see a new Email action is getting displayed with the same label.
- Label - Email
- Name - Case.Email
- Action Type - Email
- Use that Email Action and you will see like this: