The mains electric wiring & the mains to a PC's PSU power cables have hot, neutral & ground wires so connecting to the ground line should be the perfect way of allowing static charge to disperse without damaging any sensitive components.
No, your right, the ground WILL and is a great path for electricity to flow to. It will equalize your computer's potential to EARTH. However, if you are not grounded to the SAME EARTH ground then your potential is still there. You will STILL need to touch the case before touching something, which is why I call BS.
If you were to touch a electrically grounded computer's MOBO microchip or memory, the electricity doesn't magically find ground, it goes through the component, and then to ground... just as it would when ungrounded. Grounding is more important for proper operation of the computer by lessening or preventing electrical noise and harmonics rather than protection from static electricity from external sources. Grounding also drains static electricity that the computer may generate.
In fact, it could be very loosely argued that a grounded computer could suffer from more damage if grounded! The reason?
If you had a a static charge of 20,000v, and the computer had a potential of 2000v, you would have an 18,000v difference in potential (the shock).
If the computer is grounded, you essentially have a 0v potential computer, but you still have 20,000v difference of potential.
So the ungrounded system in this hypothetical situation gets "shocked less".
It should also be noted that the Neutral wire IS grounded, but is not THE ground. Your panel or breaker box at the point of service will have the neutral tied to the ground.