Hi,
Thank you for sharing more information related to the issue.
The issue seems a bit strange since emails should be logged. We suspect there might be some issue with some other plugin or theme. Performing a conflict test will be helpful in ruling out this possibility.
Will it be possible for you to perform a conflict test on your site? You may need to temporarily disable all other plugins except the Email Subscribers plugin and switch to a default WordPress theme.
We recommend doing the troubleshooting using the Health Check plugin which disables all the plugins and switches the current theme to the default theme only for the current logged-in user when troubleshooting mode is enabled. It does not affect other users or site visitors. You can get the Health Check plugin from here. After enabling the troubleshooting mode in Health Check plugin, activate the Email Subscribers plugin, and check if the issue is resolved or not.
If it is resolved, then you can activate plugins one by one or switch to the current theme and check which is causing the issue.
If the issue persists with only Email Subscribers active, do let us know if this is the case. We may then need to debug it further.
Hope this helps you.
Let us know what comes out in your testing. Also, let us know if you have any further queries.
Thank you.