first off, as a ren fest veteran, it's easier to just take "renaissance" to mean "basically any time between the collapse of rome and world war i, also with the possibility of time travellers and things that never existed in the first place and anachronisms like turkeys in pre-Colombian England" and take "Renaissance" with the capital "r" to mean "from the 1300s to roughly the discovery of the new world"
Second, I think it's an interesting question to ask if people craved this particular past or if they just craved a space where everything is vastly different and settled on this period because it's pretty firmly embedded in our cultural mythology and everyone has an idea, however historically inaccurate, of what this time period was like. I think it's less that we built it because this is specifically what we want and more that this was waiting for us to materialize it. The majority of what people get from the ren fest isn't renaissance specific, it's just about going to a place that's different enough to be novel and exciting but familiar enough to be comfortable. I think that if we did a Flintstone Fest where we all dressed as cave folk that would be similar enough that you have people who go to both. In fact I may be a little biased because I actually like to bring a caveman character to our local fest's time traveller's weekend. It's fun to turn the conceit on its head by walking around acting amazed at how advanced everything is. I could also foresee a successful Old West Fest. All time periods that are vastly different than now, but with a familiarity from being studied, stereotyped and riffed on in the common culture. A comfortable elsewhere and elsewhen.