I remember watching that very well. My parents were aghast at the screaming girls in the audience and openly scorned the whole thing. But they never turned it off either haha. I didn't get the craziness and fandom myself (I was the ripe old age of 6) but I just wanted to see and hear for myself these four English "mop tops" that were all over the radio with "She Loves You". The next day at school, my oldest brother and 3 of his friends went to class with black jeans and t-shirts, hair combed down over their eyes and sunglasses on. At that time, that was total open rebellion and created quite the stir. Their teacher couldn't do anything about the clothes and hair, but "just kindly take off the stupid glasses before we begin."
I never knew about their appearance at The Cavern Club a couple years earlier until much later when some of the Beatles early history was made known. That was before Ringo's takeover from Pete Best on the drum kit.
Would be super cool to see that Sullivan Show again and just see if it is what I remember it to be...