Hi David,
Thanks for sharing the detailed information with us.
Firstly we'd like to explain that it is not available to turn off all notifications across the whole tenant. Different notifications for Office 365, Outlook and SharePoint need to be turned on individually.
The incoming emails notification you mentioned can be created based on different conditions. For example, the email notification for the changes of SharePoint documents or items, we need to cancel the alert manually in SharePoint.
We understand what you want is to troubleshoot the cause for SharePoint performance issue(site running slow, page loading slow), we will try our best to help you.
To narrow down the issue, we suggest you try following:
1.find a page in SharePoint online which load slowly, you can find one which has the biggest performance issue.
2.follow the steps as I shared with you in
private message to collect the fiddler logs. We need the fiddler log to check the details and it is helpful for our troubleshooting.
3.When a page contains a lot of web part, it will slow down the speed when loading it.
We suggest you open a problematic page in SharePoint, in the address bar, add
?Contents=1 in the end of the URL and press Enter, then go to the
Web Part Page Maintenance page as below. (Please note that this method only usable and effective when you are using classic experience)

We suggest you check all web part you have added, see if there is any web part you don’t use now, if you are willing, you can delete it, then check if there is any improvement when you load the page.
4.Moreover, we suggest you refer to the suggestions in the articles and try troubleshooting the performance issue and increasing SharePoint performance:
Using Content Search Web Part instead of Content Query
Web Part to improve performance in SharePoint Online
Tune SharePoint Online performance
Best Regards,
Tina