Connecting all grounds directly to the supply's ground?
The main reason is that the PCB tracks and wires don't have zero impedance. 

If you think about it, higher currents will produce more voltage across a given impedance ( your ground wires) and this causes the grounds at these points to wander around in terms of voltage. So the point is no longer at ground which can cause noise to appear in some configurations. 

What happens if I separated all of the grounds from the components and directly connected them instead into the supply's ground?

Is this technique better than just separating the grounds into small and large signals before connecting them into the supply's ground?


This is called star grounding (although there will be some local ground connections) and was common in the days of point to point wiring. In practice you will do both of the above, as the local circuit sections will need a local ground which is then star connected to the main ground, IE you dont do it on a component to component level if that makes sense... 


Best practice is to usually have higher and lower currents connected to a single point, and have a separate ground connection for small signal sections.