Problem identified

If Send Mail setting is disabled in Global Options, then Privacy Tool Suite is no longer functioning.
There is a message on the front-end:

We're sorry, you can't submit an information request at this time.
In the same time, No NEW button appears in Administrator Panel.

It was 2 days full-time quest for me :-(
I made migration from Joomla 3 to Joomla 5 and thereby made copy of the site and disabled Send Mail option to avoid users get any notifications during the working process.
When Privacy Tool stopped working correctly, I thought about incorrect migration or permissions issue.
I had to create many copies of the site and play with different scenarios to understand the reason! And though about starting the whole changes with migration from scratch again.
It is time consuming process and I found the reason just accidentally while basically I thought it is a bug.

Proposed solution

The functionality of import/export shouldn't be limited. A user just can get a typical message about disabled Mail function as we get during the registration / any sending of messages at e-mail.

Alternatively, a message can be displayed like "Privacy Tool Suite needs Send Mail setting to be enabled to get working correctly". By other words, the user should understand connection between Privacy Tool Suite and Send Mail option.