If you were talking about the band members you would have a point. In this case it is not about members but about the band (notice the singular): "The Rolling Stones is a band" is the phrase used (albeit author used ‘are’ instead of ‘is’). To explain: ‘band’ is singular (a collective), hence we use 'is’. Whether the name of the band uses plural or singular doesn’t matter. If Bob Dylan were named Bobs Dylans you would also not refer to him as ‘they’, would you? Bob Dylan is the name of a singular person and so would Bobs Dylans be. The Rolling Stones is the name of a singular band (for the sake of keeping it simple I leave the current discussion of choice-pronouns out of this argument).

I understand the mistake because the name of the band includes the plural and this makes it counterintuitive to refer to the band in singular form, and as we all get lazy with grammar every now and then (and that includes me), it is an easy mistake to make.

Language also (d)evolves over time and rules are adapted to reflect that enough people keep making the same mistake. The mistake can then become part of generally accepted language.