Bamboo is kind of a tree in this case which is an area I know more about and I think many of these factors would apply to bamboo as well. First forests used to actually be more rare than today before humans came along. Europe and North America was covered by endless plains which were grazed by huge numbers of ruminant animals. Any tree that would try to grow would get grazed before it could grow so tall that the animals couldn’t eat it anymore. That’s why grass thrives in such an environment, it’s practically made to get grazed. Once cut it quickly grows back again. Once ancient humans came along and hunted most of the grazing animals extinct forest suddenly started spreading like crazy until almost the entirety of Europe was forested (which was then to a large extent deforested again after agriculture was invented). Grass simply can’t compete against trees for sunlight. Therefore I would expect less or none bamboo in areas with a huge grazing wild life populations. And I don’t mean animals that would eat the leaves, but the ones that would eat the new seedling.
Another thing that limits trees is moisture. In general the drier the climate the less beneficial it is to be a tree. That’s because deep roots are of no benefit in dry climates (but they are of huge benefit in humid climates during drought). Grass which generally have very shallow roots suck up all the rain before it can penetrate deep into the soil while deep tree roots never get any significant amount of water. Trees handle drought well but constant dryness is very detrimental for them. Dry areas also tend to have wild fires which also hamper trees. It’s simply better to be a grass (if moderately dry) or a cactus like plant if it’s extremely dry.
Another factor is soil conditions. Now I don’t know what soil bamboo prefers but I doubt it’s all soil. Soil can have huge impacts on things like pH and water availability. For example in far Northern Europe where I live you can tell that you are standing on sandy soil if all around you are spruces and pines. If you however see lots of leaf trees you are probably on a silt and clay soil. This is because conifers handle both dry and sour soil better than the local leaf trees which leads to more conifers on sand. Bamboo is probably also limited to a certain soil condition.
I hope that can at least help you develop more theories on why bamboo is not everywhere. Something important to remember is that just because a plant can grow well in a certain location doesn’t mean it will be found there. That’s because plants are always in fierce competition. I bet bamboo if intentionally planted and cared for could thrive in lots more places than its found naturally, but it just happens to not be the best plant in that location, meaning it’s outcompeted over long time scales.
Trees are not a taxonomic group. It’s rather a description of characteristics the most important of which is having a woody trunk. For example there are tree legumes and non-tree legumes. A species of tree can therefore be more closely related to a non-tree than to other trees. However it’s totally true bamboo is not a tree. A grass could in theory however hit all the characteristics that are required to be tree and would then be considered as such, however no such grass happens to exist.