They have explained the process to you tho - they can't take everyone so they select the kids at random. Of course you are disappointed as your son wanted to do it - I get that.
But unfortunately his name wasn't chosen. That's nobody's fault. It's a bit like being angry at the weather....
Staff volunteer massive amounts of their time to run D.of.E, all unpaid. Surely you can't be angry that more staff aren't giving up their time for free & why should they? They have families and lives too.
It's disappointing of course. But just just one of those things.
My school can only offer it to a section of the year group, so kids who want to do it apply, get references from staff, and there is an interview process I believe (I am not personally involved). Perhaps this is fairer than a lottery, but it also takes up time, time which staff are volunteering for free in their lunch times, after school etc. It's not reasonable to demand that the staff at your son's school do this.
Also, why shouldn't the school advertise it on their website - it's true that they offer it. Unless they state on the website that places are available for all kids, they aren't lying.
As I said, I get why you are upset, of course. But anger about this is inappropriate - nobody has done anything wrong.
What do you actually think should happen now? What outcome are you hoping for?