Why are kings almost always redheads in cartoons?

The most well-known British King, Henry VIII, was a large bulky man with a bushy red beard, dressed exactly as you imagine a King would be.

I imagine he's something of a blueprint for the appearances of fictional kings.

I have never made that sort of connection. Fictional kings? IRL kings? Sacramento Kings?Can you give some examples?

Because none of the other characters are? It’s a quick way to set them apart at a glance, I reckon.

The cartoons I've seen that always had a fat, red-haired king character were very obviously making reference to Henry VIII, who was a king, he was fat, had red hair, and did some silly things.

I've never noticed such a pattern myself and I'm not sure I'd agree that most kings are depicted as red-headed. It would be a little odd considering the relative rarity of red hair in people. What specific depictions are you talking about? Could you give us a list of examples? If you google "cartoon king", you'll find only a few redheads among dozens of brown or white-haired kings, which is what I'd expected to find. Maybe if this is a legit trend you see, it could be regional thing? Are there many red-headed people in your country?

I can't say I've noticed this, but depending on the design, it might just be about ensuring contrast of different elements. Red contrasts a yellow crown and beige skin pretty well whereas blonde hair would blend with skin and crown, and brown isn't as bad but isn't great and doesn't work with brown skin. Black works, but can be hard to do with any detail in a minimalist style or one using black outlines. This leaves, red, and grey (with grey being them more common option in my experience).

my mental image is brown hair

Because it goes with the red-black color scheme of the cards?

Symbolism and scarcity.

As for symbolism, red is a powerful color that symbolizes power, passion, danger and bloodlust. There's also a hidden, metaphysical/spiritual/archetypal symbolism: a darkened red is the first color we "perceive" as we're babies inside the womb, as the red light can reach the human tissues deeper than other light waves. Speaking of light waves, red is the longest visible wavelength, which kinda of seamlessly blends with infrared, invisible to our eyes, but can be seen by some predatory animals such as snakes (which adds to this symbolism). We can also note how things are "redshifting" throughout the entire fabric of spacetime: as the universe expansion accelerates, light becomes more and more incapable of reaching far, thus leading to a redshifting of distant stars, galaxies, nebulas and other distant space stuff. Billions of years from now, the expansion is theorized to be so fast that individual atoms will break apart, an ultimate cosmic event that is called as "Big Rip". If this were really to happen and there were still human beings alive, we'd start to see an ever-approaching redshifting, starting at the star we're orbiting (because our current Sun would be already "dead" by then, so would be the Earth and the current solar system) until eventually a redshifting horizon and, ultimately, redshifting arms and hands as every biological form still alive would start dying. Phew! I digressed... But you got the point. "Redness" can be said as something intrinsically woven within the entire fabric of spacetime, so outside our control, so powerful than anything, so red is simultaneously power, beauty, passion and danger.

As for scarcity, IIRC, the redhairness is a characteristic caused by recessive genes. So it's not as common as brown, dark or blonde hair, therefore, it can be seen as a symbol of uniqueness. Nobleness and royalty implies a sense of power and uniqueness, so it makes sense to depict queens and kings as redhaired.

Together, they sum inside our "collective unconsciousness" to see red (as well as purple, which symbolizes the other side of visible spectrum, blended with ultraviolet, also invisible to us but visible to certain animals) as a fundamental color for powerful figures.

Do they? Most cartoon kings I can think of have either white or gray hair.

If I go to Google image search and put in "cartoon king", there appear to be a mix of blonde, black, red and white haired kings. Red doesn't appear to be over-represented here. What cartoons are you thinking about?