I know the argument that criminals are still going to obtain firearms illegally, but isn't it true that a significant amount of mass shootings in the United States committed by people who obtained said firearms legally?

No. Most mass shooters are obtaining guns illegally. Most violent gun crime, in fact, is committed with guns that are obtained illegally. The ovewhelming vast majority of gun owners are perfectly law-abding.

And even if it does stop people from obtaining guns, it's not going to stop the problem. The United States is a very violent nation. People are getting stabbed, beaten up, driven into on almost a dailiy basis. The problem here is the people, not the methods of violence. Take Iceland as an example. They have one of the highest rates of gun ownership per capita in the world and yet they basically have no violent crime. In fact, the first murder to take place in Iceland in 30 years wasn't even committed with a gun. Iceland has something that the United States does not: a decent healthcare system (including mental healthcare), a decent education system, a decent criminal justice system (they were the ONLY country to jail their bankers in the wake of the 2008 financial crisis), and a decent economy.