Needing to use a staging site for testing client site updates is an essential practice. You do not want to send customers any site emails by accident since the site that has a staging site for testing had a membership or affiliate plugin active.
To make sure emails are disabled on a staging site, use the Disable Emails plugin. The plugin has no settings and should disable site email on the staging site in most cases. One extra step to remember is if you have any SMTP or transactional email plugins active on the site. Remember to deactivate those on the staging site.
As site emails will not be sent on the staging site, you will need to test different plugins that would need to send out emails for different reasons. For example, if you have an e-commerce plugin active such on your staging site like WooCommerce or Easy Digital Download and you still need to be able to test what is being sent in site emails you can install a plugin like Log Emails, which works with the Disable Emails plugin.
With the log email plugin installed and activated, you can view the log of the site emails sent from;
Tools > Log Emails
If you need to change the log retention period, those can be changed by going to the settings screen. The default is 30 days for email retention.
Settings > Log Emails
If you needed to purge the email log entries, you can do so from the tools screen. Select the number of entries to be deleted.
Having the ability to disable emails and log emails easily on your staging site means you have peace of mind knowing that site emails will not be sent out, but also to check that site emails for testing get sent out safely.