That's a damned good explanation of Kant. The point I would disagree with is whether or not the maxim-method is really that good/preferrable.
While I am by far opposed to Utilitarianism (I much prefer a Rawlsian contract theory myself), and I absolutely agree that the maxim "the sacrifice of innocent lives is moral if it saves other innocent lives" would lead to horrendous outcomes (things like organ harvesting)...
I would argue that the maxim "the sacrifice of innocent lives is moral if it saves other innocent lives" and in general the process of creating maxims out of complex situations goes too far with its simplification and loses important nuance.
This is evident by the fact the vast majority of people have a difference in their intuitions for either part of the trolley problem. It may be a logical fallacy to say so, but if there weren't a meaningful difference between the two trolley problems, why would so many people express a difference in intutions between the two problems? And when the maxim is created that nuance is lost.
To go into what I mean:
You could come up with a maxim "do not cause unnecessary harm to others."
But what about unnecessary but helpful harm (like dentistry - hardly necessary, more a matter of comfort and aesthetics for some dental problems). So the maxim is expanded to "Do not cause unnecessary harm to others unless they ask for it."
But what about someone who is mentally ill who asks you to remove a leg they think is controlled by Satan? Then the maxim is expanded to "Do not cause unnecessary harm to others unless they ask of it while of a sound mind."
But what if the person is perfectly sane, but due to cultural beliefs they sincerely and of a sound mind believe that their leg is infected with something and must be removed after touching some taboo/forbidden substance? You know via medical tests they are physically fine. But of a sound mind, they disagree. Then the maxim becomes "Do not cause unnecessary harm to others unless they ask of it while of a sound mind and the request is based in reality."
At some point the Maxim becomes complex and cumbersome and you end up with as many maxims as their are ethical problems to begin with.