Yea … kinda … okay not really. What you term "justice" in your head is what could be called punitive justice, or the seeking of a just outcome through the use of punishment.Any reference to Justice that includes an adjective is no Justice at all. There are no flavors or shades to Justice. Justice either is or isn't. What they're really saying when they want their version of "justice" is that they want an outcome that matches their narrative. It doesn't work that way. Sometimes what's just doesn't feel fair. People are going to feel let down on both sides from time to time. It helps us hold ourselves to a higher standard than we otherwise would.
As a society, we collectively agree on the idea of a crime, the criteria needed to be guilty of such, and the repercussions that should follow. Some people have a hard on for punishment. Others have different ideas on what will lead to behavioral change and how to get there and, as a result, an improved society.
For example, offering GED courses or Associate Degree classes, detox, and mental health services to prisoners while they are in the system are all restorative justice techniques, as opposed to punitive justice like solitary confinement or hard labor.
Whether you think the philosophy of restorative justice has any merit or not is up to you - but it's definitely a real thing striving for the same thing punitive justice does: a change in behavior leading to lesser recidivism and societal betterment by turning an individual societal leach into a contributing member.