The datasheet for the 2N2369 transistor and for every other American silicon transistor show that they saturate well when their base current is 1/10th the collector current. Beta has nothing to do with saturation. The base must be forced for the transistor to saturate.
You forgot to say what is the collector current so we don't know if your base current of 1mA is enough.
You forgot to say what is the collector current so we don't know if your base current of 1mA is enough.