A common misconception is that Buddhism is not a violent religion but bear in mind that some eastern empires/kingdoms were mostly Buddhist yet still found time to wage war. In japan the ikko ikki followed a radical Buddhist teaching and fought against the daimyos.

Also important is that Buddhism is often syncretic with others faiths eg Hinduism or Shintoism. Samurai for instance still venerated ancestors and kami yet stopped live fire exercises on dogs for fear of it harming their dharma.

A nice way to phrase it is “Christianity is not violent; people are violent”. The same applies to most religions and beliefs.

As for the movie there was a lot of persecution of Christians in Japan, probably due to the perceived threat to the traditional power/social structure of japan eg. Converts to Christianity often took Biblical or western names.

The Shimabara rebellion also didn’t help. It may have been due to taxes but was perceived by many as a Christian revolt in the relatively peaceful Tokugawa shogunate period.

Buddhism absolutely was spread and maintained via violence at times. You can read about it in the academic book zen at war. During world war II, nearly every zen monastery supported imperial japan and the axis powers, considering their war to be a holy war for the establishment and preservation of superior ideals. The idea of buddhism as an insanely peaceful religion is based in a large way on western stereotypes that draw mainly from descriptions of how monks act. But which were precepts that non monks were not expected to follow. And world war II isn't far back into history like the crusades. This is recent enough that there are people in those monasteries still alive today for whom this is living memory.