Sorry if any of this has been covered by posts in the six pages I haven't read, but a few thoughts from me...
I'm a regular with Let's Ride (guided and social rides), and I know a couple of my local Breeze leaders (though for genetic reasons I can't join their rides). I also have a fair bit of experience of software and web stuff, and I think the problems you describe are really because the site software is very simple and intended to be low-maintenance. It doesn't look like it sets cookies, and you can't stay logged in... the site software is primitive, to be sure, but I think that is the problem rather than any intentional bureaucracy.
But as for limiting the number of people a single leader can be responsible for, no, I very much disagree that that is "unnecessarily bureaucratic". If one leader was allowed to lead 50 people and there was an accident, you can be damn sure there would be people complaining!
I think Let's Ride is a great free resource, and the ride leaders I know are dedicated individuals who put a lot of personal effort into helping get people out on bikes. And they get almost nothing material in return - just a "Ride Leader" jersey and some discounts, I think.
And I actually find it a very easy resource to use, site wrinkles notwithstanding - it's greatly contributed to my cycling miles this year, and it's found me a new social circle.
I suggest you add your favourite leaders to your list of friends, and perhaps join any groups they're leading, and you should be able to keep in touch with oversubscribed rides easily enough. I've done that, and I don't really miss anything.