Is there a name for this type of pop song rhythm?

Cafe Society
1

A number of very popular rock songs from the 60s and 70s started off in a similar way: a repeated one-note motif, accompanied - in varying degrees - by a syncopated shuffle, often with an implied emphasis on the “and” of the first beat.

The Turtles - Elenore

Three Dog Night - One Is The Lonelist Number

The Cowsills - The Rain, The Park, And Other Things

Queen - Killer Queen

Is there a formal musical term for this pattern?

3

I’m not sure it has any special name. It’s not really a one-note motif so much as banging out the chords on each beat. As for that style of intro – I’ve never heard it referred to as anything specific but, yes, it’s not uncommon.