I'm creating this post for future references. A while ago I got bored and decided to do a design exercise on this, back then I called it 'Push Button Selector' because I did not know of the term 'Radio Push Button', and hence I could not do no research on it. I had to start from scratch - https://forum.allaboutcircuits.com/threads/push-button-selector-switch-no-ics.161832/ - Months later, reappears in - https://forum.allaboutcircuits.com/threads/sr-latch-within-an-sr-latch.168202/ - This time, Mr Crutshow has simplified it significantly and I think it is as good and simple as it could get, here is the circuit, slightly adjusted to ensure the saturation of all NPNs: