Sadly 'walking' doesn't always just mean that when in connection with the Ridgeway - it also means parking.
Although the Swindon - Devizes bus can be used to access the Ridgeway (e.g. at Barbury) without a car.
Although I only really know the Uffington to Avebury section of the Ridgeway, I would say it is challenging getting up onto the Ridgeway by bus as it always involves an uphill walk. There is a bus from Swindon to Hungerford which goes through Ashbury where there is an uphill path which will take you near to Waylands Smithy and Uffington Hillfort. Barbury Castle is difficult ... there is a rare bus which serves a housing development built on the site of what was the RAF Princess Alexandra Hospital then an uphill walk or otherwise a much longer walk up from Wroughton, the Swindon to Devizes bus (49) does go through Wroughton.
One can of course access the Ridgeway without parking nearby, but from kite fliers to dog walkers heaps of folk want to get there to do something, not do something to get there (if you know what I mean). Judged (unfairly perhaps) by the roadside parking adjacent to Adam's Grave on a Sunday morning (not all are 'walkers'), the state of the verge, and the impact on monuments (an example being Adam's Grave from people walking and in one case at least riding over it); parking is just one of the impacts that has to be considered.
Re modern walking pilgrimages what time of year is likely? Are they seasoned or inexperienced walkers? Do they know how quickly the weather can turn on the Downs? There is a lot to consider in this area. Some works I would probably look up would include books and pamphlets by Ken Watts and Adam Stout, both know the monuments very well, the former knows the Downs and routes like the back of his hand, and Adam has actually undertaken a long distance pilgrimage and did so in part study in relation to a historical theme.