I think the rule to leave your bike along the road makes sense. People walking or riding through the forest or trails (and yes, some cyclists would ride their bikes through the open forest and trails) would have an adverse impact on the forest, as well as adversely impact any hikers present.
I've toured and camped in numerous US national parks, and biked to/within a bunch of other national parks without camping in them (including Crater Lake NP). I believe that the forests at Crater Lake NP are wilderness, and forests at that elevation and latitude are somewhat fragile. Much of the national forest land around Crater Lake NP are not designated as wilderness and have logging, but there are proposals to change that.