Most particle accelerators are unable to steer two particle beams to crash head-on into one another. This is not a hard fact to understand, and you actually point it out that several times, but somehow the simple statement in so many words triggers outrage from you. Selected quotes from you:
Probably by count most accelerators were either not intended to generate collisions at all or were fixed target accelerator
Right there you just agreed that by count, most particle accelerators are unable to steer two particle beams to crash head-on into one another. End of story: you already agreed with the statement you were disagreeing with.
In either case there was only a single beam so talk about steering two beams is foolish.
And, since most particle accelerators are single beam, they are unable to steer two particle beams to crash head-on into one another. Given that you just gave explanations for why most particle accelerators are unable to steer two particle beams to crash head-on into one another, I can't see why you are vehemently objecting to the statement.