Hi Jascdad,
Since this thread has been lasted for over a month. To help us understand your current situation better, I will try making a summary. Please help us point out the misunderstanding part.
In my opinion, your current issue is about the error indicator show up continually. If you uncheck "Turn on background error checking", the error indicators will temporary disappear, but will come back after your reboot
your Mac.

If this is your situation, I would like to ask you to check if "Turn on background error checking" is enabled automatically every time you reboot your Mac. If so, then your issue should be "Excel
Preferences won't save".
Per my research, I found a user with similar issue: Problem: Excel Preferences settings keeps reverting to default.
This user figure it out by creating a new user profile on his Mac. Please try this method and check if it helps in your scenario.
Disclaimer: Microsoft provides no assurances and/or warranties, implied or otherwise, and is not responsible for the information you receive
from the third-party linked sites or any support related to technology.
If you have checked that "Turn on background error checking" remain uncheck after reboot your Mac, but the error indicator keep showing up, then, like I mentioned in
my former reply, it's might be a file related issue, if it's convenient for you, we would like a copy of this Workbook via
Private Message, so we can further investigate on this issue.
Regards,
Alex Chen