The 70-200 line of Canon lenses is among the best zoom lenses made. The 70-200 you were looking at is actually the cheapest and lightest of the series. There is also an f/4 IS (image stabilized), an f/2.8 (which I own and love), and an f/2.8 IS. The prices of these go at $1100, $1200, and $1600 respectively.
You're still thinking about big lenses meaning bigger zoom. This is actually rarely the case. The bigger the lens, the more light it will let it (ie- larger aperture (smaller f number)). I think my 24-70 is actually bigger than the 75-300. Cheap consumer zooms such as the 75-300, are smaller because they are "slower", allowing less light than some of the faster lenses.
Your next possible question would be why would you need a lens that costs more, covers less of the zoom range, and is bulkier. Fast glass (especially when you get into the 2.8 zooms and any primes) allow you to hand-hold shots at dimmer light conditions, have better bokeh (background blur), are built like tanks (think- no plastic pieces), have less optical defects (less pincushion, chromatic, etc. aberations), are virtually dust proof, and produce sharper pictures. To some, myself included, that's worth having a shorter (focal length-wise) and heavier lens, that I won't grow out of.