Why would some military aerobatic team use fighters unlike the many others that use trainer aircraft?

Budget, I suppose. Fighter airplanes are very expensive to operate, so the air force needs a lot of money to even think of allocating some of their precious fighters for aerial display. Obviously, the US military has no issues on this front, but many European display teams use jet trainers (the BAe Hawk is a popular choice around the world).

I believe only the Russians, Chinese, Singaporeans, Malaysians, and Greeks use fighters for their demo team. However, most helicopter display teams use “frontline” helos.