The criteriums during Tulsa Tough and races like Tour de France, Tour of California or The Joe Martin Stage Race are a different category of races. Somebody else will have to comment on how eligibility for those work.
But for big road rides like those that happen during Tulsa Tough or last weekends Hotter N' Hell 100 in Texas anybody can enter much like The Tulsa Run. I can personally attest to having been passed by several people riding $200 mt. bikes while riding my $1000 road bike on such rides. Most big road rides have multiple lengths as well. 100 miles usually being the top (which I suppose is roughly equivalent to a marathon?) , down to 25 or sometimes less miles with lots of options in between (sort of like the 5ks and fun runs). See
http://www.tulsatough.com/site/sections/16 for the distances and courses for Tulsa Tough.
I agree, riding has a higher barrier to entrance then running because of the gear...mainly a bike and helmet. But past the bike a lot of the gear is semi optional or optional depending on preference and desired comfort.
Overall I have found biking to be a welcoming group in my on and off recreational riding career. The vast majority of people riding in big recreational road rides probably match up with the people running in equivalent road races in terms of training, motivation, and level of seriousness.
You just buy a USAC annual license and you can start crit racing. Heck you don’t even have to do that, you can buy a one day license the day of the event even if you have never raced before. Joe Martin stage race is no different for Cat 5 racers (beginners class). Then you accumulate points based on your finishes and you move up through the categories as your skills and finishes improve. It goes up through Cat 1, then Pro.
One of the reasons I quit crit racing was the lack of any sort of requirement for bike handling skills. As you know, there’s some pretty sketchy riders on the road. Just imagine them hurtling into a corner at high speeds in the middle of a pack of other bikes.
As far as the big tour events like US Pro Challenge, TDF, Tour of California, etc. those are restricted to pro riders only, they are invitationals by team, and the pecking order on the teams is pretty much determined by the teams.