Lots of people do cover songs. The Staple Singers did a great version of The Weight by The Band. No one however has ever done a decent cover of I’ll Take You There, one of the Staple Singers greatest hits.
Some things people have the good sense not to try.
Part of the problem with covering songs is that it’s really difficult to find the mood of that period. Most of the time you’re better off just doing it your way.
Brian Setzer is a bit of an exception.
His cover of Louis Prima’s hit “Jump, Jive and Wail” is actually reasonably faithful to the original and mostly successfully so.