If a cyclist can ride right through this, why can’t I on my 125 motorcycle?
Perhaps it should be allowed! Cars already treat stop signs as yields ("California Roll" is the car corollary to the "Idaho Stop"). Why would you stop if the car behind you isn't planning to? (I'd love to see motorbike studies on this; please link me to some if you know any.)
Studies have shown that cyclists treating stop signs as yield signs leads to fewer accidents, both with cars and pedestrians.
Yielding also decreases time spent in the intersection. You have a motor underneath you. Cyclists don't. Clearing the intersection quickly prevents cross-traffic from splatting you. That's why slowing down, checking for traffic, but not stopping is so important for momentum vehicles.
The NHTSA (the US road safety org for my Canadian friends) has a good two-pager overview. It's a good place to start if you're still curious about the reasoning behind the Idaho stop.
Motorists would make a tantrum regardless, and they don't stop too! Rolling through is very common; indeed expected in many areas. After all, how is a town supposed to enforce all of their intersections? For example, San Francisco has 18,525 intersections and 2,140 officers. Assigning 10 intersections per officer and to ignore all other police needs would be insanity. They're trying their best but it's an uphill battle.