Why does only Hot Rod get magically reformatted by the Matrix of Leadership?

TL;DR - You can "have" the Matrix without "having" the Matrix.

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Some of this answer is, by necessity, based on speculation. However, this is an educated guess based on the G1 cartoon continuity.

Much of what you describe can be explained by differentiating between "possessing" the Matrix of Leadership and "accepting" the Matrix of Leadership. The Matrix is not just a physical object, it's a symbol of the burden of leadership. It contains the collected wisdom of all previous bearers, and apparently can only be opened by one who is willing to accept that wisdom and all of the responsibility it entails. When that occurs, the bearer obtains full access to the power of the Matrix, which apparently includes being upgraded into a more powerful form. Here's a rundown:

First, Optimus Prime himself. The MAN.

We don't know the full measure of Optimus Prime's acceptance, but since he remained the Autobots' leader for several million years, we have to assume that he fully accepted the burden of leadership. As you stated, it wasn't clear if Alpha Trion built Optimus' upgraded body, or if the body he built was upgraded by the Matrix.

So, let's look at Ultra Magnus:

Hot Rod/Rodimus Prime is a little trickier:

Galvatron and Scourge may have possessed the Matrix, but without accepting it there is no access to its full power (which includes the transformation/upgrade part). The side effects they experienced could be explained as the Matrix "leaking" some of its power.

This is the only explanation I've heard in 20 years that seems to make sense. Otherwise, we just have to explain each robot's upgraded status (or lack thereof) as continuity errors.

Here's an unpopular idea, which I think explains this.

Hot Rod was the Chosen One^TM.

If you look specifically at the events of Transformers: The Movie, you see Hot Rod is the protagonist. During the film, he shows a preternatural connection to The Matrix of Leadership. Eventually he is the one to use it to 'light our darkest hour'. I think this explains why he alone gets an upgrade. He's the only one who has a connection to the Matrix.

Optimus Prime's a cool dude, and a great leader, however he was merely carrying the Matrix. He didn't wield its special power like Hot Rod did. Same with Ultra Magnus (excepting the coolness and excellent leadership). Lacking a proper connection to the Matrix and ability to wield its power, they do not get upgrades.

Optimus Prime and Ultra Magnus were already "adults."

Hot Rod was "a kid" until he was matured by The Matrix.

The reason why Hot Rod was chosen was because he was the first to touch The Matrix when Optimus Prime dropped it. It is as simple as that.

If you look at the movie from 1986, it shows the ancestors that carried the matrix of leadership and the first one looked exactly like Hot Rod, the matrix chose him because it was his, thats why he transformed into Rodimus Prime, thats why he was able to open the matrix,

The reason why Hot Rod was transformed by the matrix is because he was chosen by it and his body was empowered by its energies such as orion pax was reformatted by it when it was given to him by alpha trion. what a lot of people fail to recall is that alpha trion himself is a prime and had been in orion pax's life for some time. He probaly just chose orion or the matrix did. while it is not mentioned that alpha gave orion the matrix, it also isn't mentioned that the matrix of leadership event exists for two years worth of cartoon continuity. In fact it isn't revealed until the movie. Subsequently magnus was not changed by it because he was not chosen. whether it had to do with his issues of self doubt is unknown.

Hot Rod is instrumental in fulfilling the Matrix of Leadership's intended purpose

The Matrix of Leadership has a symbolic role for the Autobots but also hidden powers of an unknown extent.

During the 1986 Movie Optimus Prime prophesies:

... but one day an Autobot shall rise from our ranks and use the power of the Matrix to light... our darkest hour.

When Hot Rod is transfigured by the Matrix at the close of the movie the same words are heard:

Now light our darkest hour!

The destruction of Unicron fulfils the 'light our darkest hour' part of the prophecy, but Hot Rod's transformation to Rodimus Prime is needed to fulfil the 'rise from the ranks' part. Hot Rod is not equipped - as he is - to lead the Autobots into a new age of peace, but as Kup observes:

I knew you had potential, lad.

This potential is used by the Matrix to create a leader who can create a new and peaceful future post-Unicron:

Let this mark the end of the Cybertronian Wars as we march forward to a new age of peace and happiness!

Now everyone knows that Hot Rod was the Chosen One after Optimus spoke to him after he became a Prime using the power of the matrix. Ultra Magnus said it himself, he's only a soldier. However, Hot Rod didn't believe that anyone was just a soldier. He believed that once the war was over, there would be no Autobots or Decepticons, just Cybertron and Cybertronians. And he was willing to give up everything about him before he became a Prime, so that the Autobots could survive the war and live their own lives after it. Plus, he showed signs of leadership on Quintessa.

Also, all you bozos missed the most important thing that deemed him worthy. For instance, he rushed into battle to save Kup from that robotic sea monster! And he fixed Kup, even though he wasn't a doctor. He used the Matrix to save everyone, Autobots and Decepticons alike, even the Decepticons on Cybertron!!! Ultra Magnus tried to use it to save himself, he didn't use it in an even darker hour, like Hot Rod. That proves that Hot Rod was worthy of the Matrix's full power.

Also, look at this picture!

Ultra Magnus is covered in armor. So when he opened up his chest, he actually opened a hole for the Matrix that was in his armor. In fact, the opening was hollow when he took it out. That's why he died when Galvatron blasted his chest.

You don't believe me, then you don't make any sense! Unless if you have a better explanation. WHICH THERE ISN'T! #Transformers=Life!

The answer simple if you've read the old marvel comics. The planet cybertron is primus, brother of unicorn whom realised battling unicorn in his original form was destroying the universe so becomes cybertron and made the transformers race to combat unicron. Primus face in the comics is the spitting image of hot rod, the chosen one properly the reincarnation of primus and unicron (in the comics) named rodimus primus. The matrix meant to be the essence of primus so hot rod will be more in tuned with it than anyone else.:-)

The Matrix adapts according to the bearer. The Matrix gave Optimus Prime Strength, Power, Wisdom without changing him physically due to his larger size from the start. With Hot Rod being smaller Autobot the Matrix had to "Modify" him not just to accommodate the power but the look of a leader that Optimus already had.

All of the points made are great, however, in Season 3, in the episode "Rise of the Primitives" all of the primitive based bots - from both sides are called by "The Assistant".

This entity has called them to stop the Creator from destroying the universe. The Creator is later known to be the original creator of Unicron and in the episode is busy with his new energy based monster named Taunatron.

In this episode there is a scene where The Assistant relives his story to the primitives and shows Unicron turning on the Creator. The Matrix is clearly seen floating away into space.

This is the only use of the Matrix and likely works in conjunction with an Autobot in proximity of Unicron - which only falls to Hot Rod in the movie.