My mom loved The Beatles and she was born 1930. She also thought they looked cute, haircuts and all. Anytime they came on the radio (and they came on the radio all the time in Germany) she'd turn it up and hum/sing along, whether it was I Want To Hold Your Hand or Come Together. The Rolling Stones otoh went over her head.
I think what made The Beatles (+ George Martin) stand out was how they soaked up wide-ranging influences (rock'n'roll, Brit Beat, rhythm & blues, country, folk, Eastern, soul/Motown, English Music Hall, classical and even a little jazz) and incorporated them in their own writing, sounding at the same time varied, yet instantly recognizable. I think coming from a harbor town (Liverpool) and learning their chops in another one (Hamburg) had a lot to do with that. They were immersed in what today we would call world music at a point in time when very few musicians their age were. Plus they had a guy with a one-of-a-kind musical ear (Paul) plus a witty lyricist (Lennon) with a musical edge. Their musical development in a few years was astounding: They started as Everly Brothers/Buddy Holly soundalikes and within a few years produced music you had never heard before.
And I still think Sgt. Pepper is a no-holds-barred work of art, not a duff track on it and a sonic experience to this day. I believe their iconic status in pop music of the last 50 to 60 years or so is entirely justified and well-deserved. If you took them out of the equation, I don't think that pop music would have been such a pillar of youth and eventually adult culture from the 60ies onwards.