Are your rates really going to skyrocket for ONE ticket in 20 years? Doubtful . . . if so, you should definitely check out other insurance companies.
I had a similar incident (not in Detroit) - I was quite pregnant and having issues with urgent urination. I was speeding (8 over on the freeway) and got stopped. The officer did follow me to the nearest place I could use the restroom but then still gave me a ticket. I was totally going to do the "admit with explanation" for my circumstances. There were 87 people at Traffic Court ahead of me (reverse alphabetical order) and EVERY SINGLE ONE OF THEM did the explanation thing. The judge listened patiently to each story, but not in one instance did he change the amount owing. When my turn came (3.5 hrs later due to so many explanations) I told the judge that I respected his time and the time of those in the courtroom far too much to come up with a lame excuse for breaking the law. I got a standing ovation and the judge zero'd my points for the ticket (but not the fine).
I know that I got lucky and doubtful this would work again. However, I decided after listening to all the bs the judge was handed hours before my turn, my excuse sounded no better than anyone else's.
I would admit guilt, pay the fine, and change insurance companies if I were you.