If the government were not enforcing existing laws against rape and murder, damn right there would be lawsuits.

If there were no laws or no enforcement of laws against such things you would certainly sue (especially in the original sense of to plead and make arguments for a point).

The US is a uniquely litigious country so going through the court system in an adversarial sense makes perfect sense. In The UK system you might instead ask a court to determine who or which state organ has responsibility to fix the problem or who has authority to do something. See for instance the judgement on the Brexit case: https://www.judiciary.gov.uk/wp-content/uploads/2016/11/judg....