In fact you are wrong. The 5th amendment applies to all and any court proceedings, affidavits, discovery motions, anything that might lead to uncovering something that MIGHT be used in a criminal case.
If you were driving down the street with a carload of stolen stuff, and you went off the road and hit a fence, you could immediately shut your mouth and demand an attorney, even if all that was going on was the fence owner wanted you to get his fence repainted.
No, you mistated it, and you are wrong as to the application of the Fifth Amendment.
It applies to self-incrimination, and does not apply to matters where criminal admissions are not an issue.
If you do not want to say something after an accident, it is in your example because the person knew they were involved in a criminal activity. the Fifth Amendment applies to that.
If a civil motion is filed against you, and you do not respond, you have defaulted and in all likelihood the motion would be decided against you.