If you use the same lens and change the enlarger height, then you change the magnification and therefore the exposure time at the same aperture will be longer at the higher magnification. So yes, if you make an 8x10" with a 135mm lens then raise the head to make an 11x14" with the same lens, aperture, and neg, of course you will need a longer exposure for the 11x14". Agreed so far.
If you change lenses and make two enlargements of the same size at the same aperture, however, enlarger height will be different, but the exposure time will be the same. So if you make an 8x10" with a 135mm lens, then switch to a 150mm lens, it will be necessary to raise the enlarger head to make an 8x10", but the exposure will be the same as with the 135mm lens, because the magnification factor is the same. It will also be the same if you raise the enlarger head way high and project on the floor to make an 8x10" with a 300mm lens from the same negative, or if you project across the room and make an 8x10" with a 1000mm lens from the same negative, because in each case, you are making an 8x10" from the same neg.
It is true that raising the enlarger head to make a bigger print without changing the lens, as you say, will result in increasing the exposure time, but if you change lenses so that you are not in fact making a bigger print by raising the enlarger head, but making the same size print, the exposure will be the same, because the magnification factor is the same.
In general, I would agree that one should use the shortest lens that will adequately cover the format, though when the differences in focal length are small, there may be particular reasons--economic, optical, and otherwise--to prefer one lens over the other.