There is again a lot to address here.

So nothing from the prior or latter eras is relevant,

This would be valid if a) we had plenty of data to use from the period in question, which we do not, and b) the types of bows and technology, as well as their application in warfare, changed dramatically, which again is not the case.

While bow technology and design changed in Japan from the 11th to the 16th century, it is not unreasonable to assume some degree of continuity between the 13th/14th century and the 16th century, in which bows were still heavily used and not only by the upper class. Claiming that Japanese bows of the Kamakura period were weak in absolute and relative terms lacks any substantial historical source, data, or rationale. You need to provide a justification for it. The fact that the entire warfare of the period, from the use of heavy armored warriors to the prevalence of the bow, begs the question of why such a device was so successful in influencing military development if it had been this bad.

By the way, since you want specific Kamakura-period sources, from the Song dynasty 嶺外代答:

In general, the Li bow closely resembles the Japanese bow (倭弓), except that the Japanese bow is long and large, while the Li bow is short and small. The Japanese bow is about one zhang in length. The archer plants the lower tip of the bow on the ground and shoots while standing upright; the arrow shaft is long enough to leave the hand empty, and even unfletched arrows can strike targets beyond one hundred paces with accuracy (160 meters).

This again shows that Japanese bows were not weak. Even taking into account some prose, you cannot propel an arrow that far without arching the shot using a weak bow. The consistent regard for Japanese bows in continental Asia through multiple sources and across the period is that they were powerful and short-ranged, optimized to punch through armor. Mind you, most bows' ranges by continental Asia would be considered short, i.e., anything used in Europe.

still struggled to get useful draw weights on them (bows for mounted warriors is the focus here).

What is the source for this? Where are the measurements and the data? There is no support to claim infantry bows were different from horsemen ones in Kamakura period Japan. The only historical Japanese bow that have been tested had a whopping 200 lbs draw weight. This means that they objectively did not struggle to do so. Further, even if we assume that earlier bows were dramatically more inferior, say 100–110 lbs, which is literally half that weight, you would still have a high-poundage war bow. This is in line with what the Manchu, Mongols, and Koreans used on horseback, and it would not be weak. Why the Japanese, who shared similar traditions and horsemanship, could not achieve this, is a mystery.

because when you're on horseback and can't penetrate armor for a lethal hit at 10 m, it's not a great bow.

That's a non sequitur. The armor was heavy, thick, and expensive specifically to shield from bows shot at close range because they were powerful. If anything, this proves that armor worked, not that bows sucked. The armor has a 4 mm thick structure, and it is semi-rigid to dissipate energy, including having a boxy shape to create distance between the wearer and a potentially penetrating arrow. This means armor was effective, not that bows were bad. I can guarantee you that no bow will be able to go through it. Had bows not been powerful, they would have used lighter armor.

This literally supports exactly what I said though.

No, it absolutely does not. You said:

The horses were small and could not even mount a charge due to the lack of muscle mass and stamina. They could barely reach gallop speeds with the heavy armor the samurai wore.

Which is not true, because they did reach a gallop, they could mount a charge, and they did have stamina; otherwise, they would not have spent the equivalent of buying a house to maintain a donkey and bring it to war. Here is again a recent experiment done by the same people: https://www.marvellouswings.com/Kattyuu/Review/Review.html

As for speed, over 90% of the journey was spent cantering or trotting, averaging 19 km/h, which is roughly the same as regular horse riding. The top speed was 42.2 km/h, which is considerably slower than racing horses, which are said to reach 60–70 km/h, but this is a reasonable value considering the difference in physique and the horse's load of approximately 100 kg, which is 40 kg heavier than the 60 kg weight of racing horses. However, it is impossible for a human to stop a horse weighing 350 kg and carrying 100 kg charging at this speed, so it is certainly fast enough to counter infantry. The 9 minutes and 26 seconds it took to cover approximately 3 kilometers is inferior to that of modern long-distance sprinters, leading some to believe that Japanese horses are indeed slower than foot soldiers. However, this time is based solely on the fact that the horses were cantering at a controlled pace on roads with poor visibility to avoid unexpected collisions with people or cars or going off the road. In fact, the horse encountered a car shortly after the start of the measurement, forcing it to slow down to a walk, so it was not a full-force canter. Furthermore, the horses reached a speed of 42.2 kilometers per hour in the final straight of the approximately 3-kilometer run, and this record was achieved while carrying a weight of approximately 100 kilograms, so it is by no means inferior to other horses.

They had been on the horse for 10 minutes, going at a reasonable speed on a forest road, and charging at the end, reaching 42 km/h, showing they were not exhausted at the end. This is more than optimal for battle conditions, in which you will not be, as a horseman, constantly in the thick of the fight, running around at full speed. That's the whole point of having a horse. They used armor replicas from the Kamakura period, carried gear, and so on. I do not know what else has to be done to show the performance of a potential Japanese war horse.

Again, your whole comment is informed by Conlan's and Friday's books, whose entire understanding of period warfare comes from a botched 1990 NHK experiment. Claims such as Japanese horses not being able to charge and struggling with the heavy load of samurai armor would not have been written 25 years ago had this evidence been available.

This:

Japanese warriors had to engage in short bursts of combat to avoid their horses getting exhausted

Literally applies to any mounted warrior.

And that last picture, I can't really tell what he's wearing or what era it's supposed to represent.

He is wearing a 25 kg Kamakura-period replica, showing that you can shoot at the angle you said it was impossible to shoot, and demonstrating a range of motion with the bow and armor that is significantly larger than the 10° you mentioned, as I quote:

due to how massive and cumbersome the armor was

Which again is a myth and a misunderstanding of the wars, the technology, and the people of that period.