Me feeling is that cyclists have a right to the path, as do people walking on it.
HOWEVER, I don't think that cyclists have an automatic right to the path which is over and above that of the people on foot. My own experience is that people walking tend not to complain about cyclists until they either get hit by one or have a very close call. The idea that those on foot should automatically "get out of the way" of cyclists (which is what I have been told on more than one occasion by somebody attempting to hurtle past me very quickly on too narrow a portion of track) seems misplaced.
If all those on the camino treated each other with an equal amount of respect, regardless whether on foot, bike or four-legged transport, I doubt that this theme would come up nearly as often as it does. Maybe I am biased, as I am not a cyclist, but there certainly does appear to be a sense of entitlement exhibited by some of those who do the camino on two wheels, that those on foot are less important that those on a bike (and hence should either make way or be pushed out of the way). By contrast, I've never been shoved out of the way or knocked of balance by either another pilgrim on foot or by one on horseback.
My view is by no means intended to imply that those on bikes should not be on the camino. Rather, that if one chooses to make the camino journey on a bike, then expect that the path ahead will have 8-10 times as many walkers on it than cyclists, and that this will mean that you may not be able to safely travel at the speeds you might otherwise want to. There are several people on the forum who travel by bike - recently one seemingly very courteous cyclist commented that it was simply too difficult to travel on the "official" camino path, for various reasons, and had elected to take the roads instead (which by all accounts are not too heavily trafficked, with the extensive new network of autovias that take the majority of motorised traffic off the smaller roads, which are now pretty good for cyclists).
If you do choose to cycle on the same path as pilgrims on foot, using a bell to announce your presence would seem a courteous thing to do, rather than approaching at speed, unannounced and giving a shove, which unfortunately on some occasions seems the only other solution considered. Or just be prepared to slow down, make conversation, and relax... I for one am happy to step aside, if the track allows space for this, and if I have enough time to do so before a cyclist is upon me. The use of a bell would allow this to happen more easily.
Alternatively, I understand that Spain (and Europe) has a good selection of trails and roads perfect for cycling, and much less encumbered by foot traffic.
As somebody has already pointed out, however, on this forum I am likely preaching to the choir...