The presence of bikes or not definitely doesn't make or break "wilderness". I've seen Wilderness areas all across the spectrum. The ones where I live now tend to have crowding issues as it is from folks who just want to hike them. A permit system to keep crowds down makes a lot of sense for crowded Wilderness areas. I see traffic levels in general as a bigger issue than what users are out there. I think what makes more sense from a management perspective is to look at trail density and use levels in the context of management goals. Frontcountry areas should permit higher trail densities and use levels. The more "backcountry" a place gets, the lower trail densities should be (with bigger distances involved) and the more land managers should consider permitting systems to limit the number of users at a given time.Maybe there is little left where you live, but out here wilderness, monuments, and national parks are everywhere.
A place like Escalante is huge. There is no reason why there could not be hundreds of miles of single track available for riding around the red rock. There are already dirt roads through the area. That does not mean MTB use would be allowed everywhere in Escalante, but that type of distance would be tiny part of the total space.
I used to live fairly close to some Wilderness areas in east TX, too. hiked in one, and the "trail" I was using was a closed county road that was still paved. I found trash there dating back to the 50's or 60's or so. I understand that there were habitat conservation reasons why ppl wanted that area protected (longleaf pine savannah restoration), but I don't think "Wilderness" was a proper land designation to accomplish that goal.
And I also spent a summer in the town of Escalante, UT working for the USFS as a wildlife biologist. There's a LOT of area there that's far more wilderness than anything I've seen in the east.