The call center isn't going to say "Nothing has started on your application. Good luck". They're obviously going to tell you something on the lines of "We've everything we need and your application is being processed".
Having said that, the Residence, Language and Physical Presence will all say "Not started" until you are close to getting a "Decision Made" on your application. Knowledge will be set to "Completed" once your test results are validated.
I completed my test on June 28th, it was validated on June 29th. In my most recent notes (extracted on 14th July 2021), I had everything set to "Not Started" except "Knowledge" and Background checks. So, this is normal and this is how the status will show up on the GCMS notes.
I received DM on August 3rd. Once I receive the next set of notes, I'll post on the September 2020 thread as to when my Residence, Language and Physical Presence were set to "In Progress" on my GCMS file to give you an idea.
Below are the posts where I posted the breakdown of my first two sets of notes. Take a look and compare my notes with your notes.
Notes before test invite =>
https://www.canadavisa.com/canada-immigration-discussion-board/threads/september-2020-citizenship-applicants.700718/post-9437895
Notes after test completion =>
https://www.canadavisa.com/canada-immigration-discussion-board/posts/9561211/
Also, about "move to another visa office" - there is no guarantee that this will speed up the application.
I hope you see some progress soon.
Wish you all the very best.