Companies don’t have to sponsor you (initially), it is possible to work in the US with an F-1 visa. Normally for 1 year (OPT), but STEM-OPT extends this to 3 years total. Only after that do you have to leave the US. And yes, F-1 is not a dual-intent visa, but H-1B is. So after graduation you get 3 years to attempt to get an H-1B, which is a dual-intent visa and allows for a Green card application. It is also true that not all companies sponsor internationals, but most tech companies do; certainly all of the big ones (Microsoft, Google, Amazon, etc.). In fact, teams there sometimes consist of only Indians/Chinese citizens even though they’re all located in the US.
Of course, there are a few roles in which one cannot work as a non-American, typically those that require a security clearance. These are indeed impossible to get, but they only make up a small percentage of the industry. Hence, a non issue really.