An "assault rifle" is specifically a selective-fire rifle with detachable magazines and intermediate cartridges. AR-15s, AK-47s, and M16s meet this definition. You are likely thinking of "assault weapon," a term which is not well-defined.
And while it's true that most mass shootings and gun deaths in general are perpetrated by handguns, assault rifles are responsible for the deadliest mass shootings.
Because it is so challenging to pass gun control legislation in the US, the least we can hope to do is forbid ownership of the deadliest types of guns.
I agree that this is not sufficient though. We need to have more stringent requirements for acquiring any firearm. 28 states don't even require background checks for private sale of guns. Our laws fall way too short on gun trafficking.
The illegal gun market is just a symptom of the very legal gun market. The head of the ATF even said, "virtually every crime gun in the US starts off as a legal firearm."
We need background checks, and I don't think private unlicensed gun sales should be legal either.