Your logic is correct, but as gonzogolf states, the lens focal length doesn't really change. What changes is the sensor behind it. Everyone referes back to how a lens would perform on "full frame", which is defined as the approx 24x36mm size image produced by film and digital cameras such as the 5D Mark II.
But let's look at it a little differently.
If you used 55-250 lens on 60D (1.6X crop) and liked the "reach" that lens gives you...
With a full frame 5DII (1.0X) you would need an 88-400mm lens to acheive the same.
Nikon crop camera (1.5X) is so similar to Canon crop (1.6X) you really wouldn't need much different than the 55-250mm.
Olympus (2X) on the other hand could give you similar reach with 44-200mm lens.
Now you might not be able to get lenses that exactly match up with the above. And, really there are many other reasons to choose one format over another. It's not only about the lenses.
"Crop sensor" such as Canon 1.6X and Nikon 1.5X are popular because they offer a reasonably nice compromise of price, size and image quality. Larger sensors (full frame) will cost more and generally will need bigger lenses to achieve the same result. Smaller sensors (2X) will struggle more to maintain good image quality and theres more of a limit to how much resolution is possible, because only a finite number of pixels can be crammed into a sensor. Smaller, more crowded sensors are more prone to noise and give less "clean" images at higher ISOs.
Camera sensors are made from a standard silicon wafer. You can make 80 sensors from one, in the 1.5x or 1.6X size (approx. 15x22mm). You can only make 20 full frame (24x36mm) sensors from the same wafer. There are flaws in silicon wafers, too. Say, for example, they average three flaws on each wafer's surface, randomly scattered but generally meaning 3 unusable sensors on each wafer. With the amaller crop sensors, that would be a loss rate of under 4%. Howwever, with the full frame size sensor, the loss rate would be much higher 15%. The even smaller "Four/Thirds" format Olympus uses would be even more cost efficient in manufacturing, but as already mentioned, the small size would be more challenging in other ways.
FYI, Canon also makes 1D series cameras with a 1.3X crop sensor. Sigma's cameras are 1.7X. Some Leica digital are 1.33X. It's a more complicated world these days! Sometimes I wish we were all still shooting 35mm film with our SLRs, where there were mostly "full frame" and a few "half frame" cameras... not many other variations. (Instead there were many other film sizes, that fit other cameras and used various other formats ranging from subminiature to large sheet film sizes.)
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