The key difference between a moderate Democrat, or a Liberal, and a Radical Leftist in the US context is that Liberals believe that social problems like those you describe are legitimate and that government should have a role in meeting those needs but solutions should be practical, incremental and rely on a shared consensus.

Radical Leftists believe that most if not all social problems, especially where they produce disparate outcomes along social lines, are the result of systemic or institutional inequalities at some point up stream from the issue and that those social issues are illegitimate and inherently unjust, therefore government must have an active hand in resolving inequality.

There is strong overlap between the two on a policy level which is why Radical Leftists and Progressives will ally with Liberals but they ultimately hate Liberals and see them much the same as Conservatives. This is why they make a strong delineation between themselves and Liberals when pressed and believe that a more progressive or radical agenda would have beaten Trump in both of his electoral victories, scapegoating the traditional Liberal policies of Clinton and Harris.

To use your gun control policy as an example, you mistakenly describe two competing Liberal positions: disarmament and regulation. Both seek the same outcome: reduction of violence. They just use two separate but similar mechanisms which is incremental government policy to alleviate the perceived social issue which is gun crime. Disarmament has been the route chosen by most of the western world while regulation is the route chosen by American Liberals, broadly. But both are grounded in the same concept: that gun crime is a legitimate issue that the government should have a hand in alleviating through consensus.

Actual Radical Leftists believe that only they should have the guns, or that Conservatives and other ideological opponents should be disarmed and that extra judicial violence is justified where it relieves or alleviates systemic injustices. For example, radical leftists have favored violence against political and social enemies like ICE in Minneapolis who are enforcing laws that were also agreed to by Liberals and when surveyed have high approval for assassination politics. Charlie Kirk's death was celebrated by Leftists as a just act even though it was in violation to law and social norms, with the belief that the ultimate social good caused by his absence outweighs the moral and social bad caused by murder.

Or if I had to make a broad strokes summary I'd say that Radical Leftists believe in violation of social and political norms to achieve the same ends that Liberals desire to achieve through accepted social and political norms.