I would suggest to first try out a user triggered AutoSupport with the subject having the word "test" in it and see if NetApp sends a confirmation e-mail back to the listed contact. In 7-mode, this would be "options autosupport.doit test". The word TEST in the subject enables the NetApp e-mail acknowledgement mechanism.
If that works, I would have high confidence that the weekly AutoSupport will work since the majority of the workings involved are duplicated with that test.
If there is something truly unique about the weekly AutoSupport then the only way I know to cause that exact behavior is to set the time on the filer forward (not backwards) to Friday night at 11:30PM local time on the filer and wait about an hour (maybe 90 minutes) for it to trigger the AutoSupport and send it.
As far as I know, the SP configuration and state doesn't control Data ONTAP's AutoSupport or impact it. The Service Processor has its own AutoSupport and its own network and uses Data ONTAP's configuration information for helping configure it (not the other way around).
Focusing on Data ONTAP is what I would do.
I would focus on the user triggered AutoSupport. If none are showing up in the now.netapp.com MyAutoSupport viewer (https://now.netapp.com/NOW/asuphome/), I would suspect a networking problem first and a configuration problem second.
For quick help, I would look in the <root volume>/etc/messages file and check out all "asup.*" lines and see if there are any issues and helpful error messages saying things about HTTP, HTTPS or POSTing.
Alas, the next steps are more difficult. In a future release, we have significant improved AutoSupport by providing a transmission history report and other commands. In 8.0, one needs to drop into diagnostic shells and do commands such as "telnet support.netapp.com 80" and see if it connects (without a banner or other response). This is a very unforgiving access level where mistakes or misunderstandings can lead to extended downtime and other issues.
A cool trick is to call the test autosupport "WEEKLY_LOG" and it will update the my autosupport site witht he latest asup too... so if you want the latest processed now instead of waiting until Sunday.
BTW, please consider plugging in your wrench port. It gives you access to e0M to directly manage the system and allows you to leverage all of the remote management and remote support capabilities of the Service Processor.