Q Why is it that I can't cast my 8-, 9- or 10-weight any farther than I can cast my 5-weight? Shouldn't I be able to cast the heavier line farther?
You should definitely be able to cast an 8-, 9- or 10-weight line farther than a 5 weight. Casting distance is mainly controlled by how fast the tip speeds up and stops over a very short distance at the end of the casting stroke. Apparently, you are not getting your tip to speed up fast enough and stop as abruptly as you should. Learning the double haul would also help.
Few people realize it, but the speed of the line hand during the double haul should be compared with a gearshift. When you need the line to travel faster/farther, then the speed of the haul should be increased. Casters mistakenly apply too much force with the rod hand.