Your lawyer would encourage you to pursue this for years, and the judges, and their descendants, would enjoy watching you go old and grey.
My take would be I'd be laughed out of court
It would take so long as to be hardly worth it, if at all. And it wouldn't surprise me if there was some way of the authorities denying responsibility.
Clearly, you have never done this and you base this only on your presumptions. Which are wrong.
It is simple to do and my two friends have done it successfuly when they had their tyres, rims and part of suspension damaged in big potholes.
First you need to have a police report on the incident, so there is a legal proof. Then you check who is in charge of maintaing the particular road (a few minutes search in the Internet). Then you contact them and file official paper on this, attaching photos, mechanical expertise of the damages and police report. It doesn't have to go to court. If it will, you'd win if you do it right. It does not take years, maybe months if the court is slow.