Buddhism condemns violence of any kind. It brings suffering to us and others.Though other predominant Buddhist countries have armies, the leader of the state isn't a spiritual leader - unlike the (13th) Dalai Lama.

There's this misunderstanding that because Buddhism often places a premium on non-violence that cuts both ways. Either there is the glossing over of episodes of violence in Tibetan or Himalayan history, or the more cynical viewpoint that just, straight up ignores teachings on violence or non-violence in Buddhism, and treats Tibet and Himalayan Buddhist countries just as any other country. I have to say I'm more partial to the latter viewpoint, since I do think it's borne out more in the historical record, while the former viewpoint has a tendency to whitewash episodes of history.

For example, Thomas Laird's The Story of Tibet: Conversations with the Dalai Lama, is a great book on the major episodes of history that Tibetans define themselves with, but it does have a tendency to ignore episodes of violence in Himalayan and Tibetan history that Tibetan agents either initiated or engaged in. There's virtually no mention of the wars with Bhutan and Ladakh, the expansion and conquests of Tri Ralpacan are glossed over, and the politicking and medieval wars between the pre-Dalai Lama Kings (the Phagmodrupa, the Rinpungpa, and the Tsangpa) are glossed over (though to be fair, this is a somewhat complex story that is seldom told outside of academic or Tibetan lit). And your post above makes a distinction between secular Kings from spiritual leaders.

While there is very much a difference in Buddhism, the spiritual power of what we might otherwise consider "secular" Kings in Buddhist countries... well, I haven't really encountered any King of a Buddhist country pre-20th Century that was regarded as 100% secular. There is very much a difference, and if we have to draw a line, it's at what point the leader takes vows of ordination, but in terms of Buddhist Kingship and Politics, and speaking specifically about the medieval Tibetan Kings, the spirituality of these kings was intertwined with their secular role. The Phagmodrupa today, centuries after their dethroning from political power in Tibet, are spiritual leaders, and when they first rose to power, it was as a monk and lieutenant of the reigning Sakya power. (And the Rinpungpa for that matter, who came after them.)

The Dalai Lama himself engages in this historical narrative. In the above book, which again, I think is a useful introduction to Tibetan history, and how Tibetans view their own history, does include the quote, "This was why Tibet was conquered: we disarmed ourselves [through adopting Buddhism]." And I don't think the historical narrative bears that out. Not in the slightest. The Phagmodrupa happily overthrew the Sakya and made themselves Kings of Tibet. The Ganden Phodrang (i.e. the Gelukpa Government started under the 5th Dalai Lama) certainly didn't reject Gushri Khaan's gift of Tibet after Lhasa was conquered in 1642. And Tibetan Armies were instrumental in throwing back the Dzungars in the 1700s, and the rise of Pholhanas that followed.

All that said, I do think that the cynical take of ignoring Buddhism's ideas regarding non-violence in regard to Buddhist history can be... too cynical. But there hasn't really been, as far as I can tell, a full attempt at understanding how Buddhist teachings of non-violence have been use throughout Tibetan history. There's been a lot written about how Buddhists, and Tibetan Lamas, have justified or managed episodes of violence in Buddhist history (see Jerryson and Juergensmeyer's Buddhist Warfare, and specifically Maher's "Sacralized Warfare: The Fifth Dalai Lama and the Discourse of Religious Violence" within). But far less about how Buddhist non-violence has influenced Tibetan warfare and history. I think it very much is deserving of a formal study, as there's plenty of contradictory evidence.

Just to take a few examples from Heinrich Harrer's Seven Years in Tibet:

• He reports that butchers are forced to live on the edges of villages, as they deal with blood and killing, and therefore are somewhat shunned in society. Evidence I can corroborate as late as 2013, having met people who rejected a marriage proposal to a butcher's son because it collected bad karma.
• He reports meeting a nomad who owned a matchlock firearm he used to try and hunt, with seemingly no hesitation.
• He also reports that he and his traveling partner Peter Aufschneiter were almost killed and robbed by a nomadic band of Khampas, as the band's women were praying and counting prayer beads to preemptively ask for forgiveness and to prepare to expunge the karma for what the men were about to do.