I have the same problem.
Example: office.outlook.com runs perfectly fine in Safari - I have set office.outlook.com to have no blockers, and lockdown mode is off - this was done using "settings for office.outlook.com" from the Safari menu. However when it is run as a Web App, it is missing many visual elements, such as icons and it becomes unusable.
The Web App was created using "Add to Dock".
I have confirmed that office.outlook.com is excluded from blockers and lockdown in Safari/Settings/Websites.
It appears that Safari Web Apps are not honouring the Website settings for blockers and lockdown.
I have attached an image of the outlook toolbar below, so you can see how useless Outlook is when running as a web app.
