This is correct, but needs to be approached with care. The gain, Beta, is not a constant. It varies as the collector current varies, and as temperature varies. The gain will reduce near saturation and it will not be the same for all transistors of the same type. Most transistor data sheets do not quote a maximum value of Beta.
If your design needs a certain value of Beta, it won't work in practice, or at minimum it will need adjustment if the transistor is changed.
Transistor circuits are often switching circuits or linear circuits.
For a switching circuit, allow for a Beta that is 5 - 10 times lower than the figure quoted for the transistor, so that the transistor will be properly saturated. You may be able to use a smaller margin if the Beta is quoted at or near saturation.
For linear circuits, make sure that the circuit will work with any value of Beta from about half what is quoted, up to infinity. In your circuit, you should allow for Ib to be anywhere from 0 to twice what it should be. As Ib is quite small, that isn't usually a problem, and calculating it gives you a feel for how much voltages will vary in practice.
For instance, if Ib is 0.1 mA, then see what Vb would be with Ib = 0 and Ib = 0.2 mA. If the voltage change is only 0.25 V on a 12 V circuit, that will probably work fine.