"maybe that's why the cowboys won? bullets vs arrows?"

Contrary to what you see in the westerns the Native Americans lost because initially they were decimated by disease.

Then we finished them off using the tactics of genocide: destroying crops and game animals, intentional biological warfare, forced relocations, and indiscriminate killing of women and children to weaken the Indian nations. The native Americans initially had no idea of how bloodthirsty we could be until they were weak and outnumbered.

In many cases the Indians had more modern and capable weapons than the US forces. The most famous case is at "Custer's last stand" where the Indian nations had modern repeating rifles that were both more accurate and faster firing than the single-shot carbines most of the US troops had. Just one of the many reasons they lost that one.

After the 1700s the native Americans had weapons about equivalent to or better than the US military. The civilian weapons they traded for were often superior to what the hide-bound military was using. But to no avail. Disease, starvation, economic and ecological destruction had already destroyed their population base.

Lesson being that survival is a system. You can't focus on physical security while neglecting other aspects. In the long run the race does not always go to the stronger, smarter, more moral or the better armed.