I'm not convinced that the question you linked to is entirely appropriate for your use case.
The accepted answer on it seems to be for circumstances where modals contain notifications which are secondary to the main task. In your situation the modal contains the primary task, so it is not distracting you from anything. From this view point, having everything in a modal would not be a distraction and would not impact usability, as there is nothing else to distract from.
Putting that to one side, using modals for primary tasks is a bit mental.
Firstly, modals can be closed by accident, potentially losing information that you have entered. You can prevent clicking outside the modal from closing it, but then, really, how is your modal any different to a separate page anyway?!
Secondly, you are toying with web conventions for no appreciable reason, which may cause users to lose trust in your system. Users cannot click any of your other navigation buttons and are not 'connected' to your main web page experience. Personally it would make me feel a bit claustrophobic!
It sounds like the site is an attempt at creating a SPA without the technical clout to properly implement this.
If I were you I would do two things:
- Try to find out why a modal is being used for the primary action instead of a dedicated page.
- Recommend that, if the reasons given for using modals are spurious, that they be replaced with proper pages (traditional, or as part of a proper SPA), in order to increase user confidence in the system.
If in doubt, test with real users.
hope that helps :)