They’re probably no better or worse than most religions, historically speaking.
Expanding a little on Japan. While religious leaders could not participate in Civil government the buddhists in the 14th-16th century would reguarly prevent the government from acting against their wishes and interests by threatening with excomunication and violence. The different sects would fight over beliefs and more often over perceived slights. The more wealthy (plunder, renting out forces, blackmail) sects grew to have influence over towns and parts of provinces and would resent interference by the civil or military authorities and they were not shy of resorting to force.
When Hideyoshi came to power one the warnings he send out was
I’m afraid it wasn’t all peace, love and understanding, but…well…you know…people like to be involved.
To add to this, Japanese Zen stories, especially Renzai, are full of violence. People cutting off body parts, banging legs in gates, etc. and becoming enlightened. These stories aren’t to be taken literally and if you read them, some are pretty funny, but still!
Anyway, I know that monks and monasteries relied on benefactors and in return, they provided training in zen. Many times zen methods were used by people (eg. Samurai) for their own violent purposes. For example, different forms of meditation can help to focus swordsmanship. Even more, it is great for dismissing fear of killing or dying.
I was not aware of monks regularly getting involved in state affairs. I was even less aware of literal battles between zen schools. Do you have anything I can read on that Saitou? In fact, many people have tried to blame zen on Japanese nationalism and militarism in the late 19th century through WWII. This is definitely not true because Emperor whatshisface dismantled organized buddhism and monks were actively persecuted. That’s part of the reason why there’s very few zen monks in Japan. They typically have a priests who get married and usually need another job to feed their family and teach. Typically, they make money by presiding over Japanese funerals.
One clear example of not very peaceful Buddhists is in Sri Lanka.
They weren’t Zen temples (they were Tendai), but there was armed conflict between the monks of Enryaku-ji and Mii-dera, and the two temples united in an attack against Kofuku-ji (which was Hossu), after Kofuki-ji monks burnt Mii-dera in 1081.
The temples of Mt. Hiei (which included both Enyaki-ji and Mii-dera) were infamous for their warrior monks and their violence until they were crushed by Nobunaga in 1571.
Add to that that a lot of the temples and monasteries got patronage from the secular leadership, and a lot of religious leaders, especially of the larger and richer temples came from buke or kuge families. So there were all sorts of personal, familial and professional ties between the civil government and the religious leadership. You even had some daimyo who were lay monks, most famously, Uesugi Kenshin and Takeda Shingen (both Kenshin and Shingen are dharma names).
The haibutsu kishaku movement under the Meiji Emperor, but that was of short duration and was ended after a few years. I don’t think many people are saying that the nationalism and militarism are due to Zen, but instead that the Zen leadership (and the Buddhist leadership in general) embraced and supported the Japanese nationalism and militarism of the late 19th century through WWII. The only Buddhist group in Japan to speak out against it was Soka-gakkai, which was why the Soka-gakkai leadership was imprisoned in 1943 and Makiguchi died in prison.