No, my understanding of this has been that it is not the musician's responsibility to do this, it is a venue's responsibility. What's a musician going to do, say, "Show me your license?" Also, all musicians having to buy licenses seems ridiculously onerous. I'll try to check elsewhere on this.

The only thing I can think of is that it could be that you are correct BUT that ASCAP/BMI, etc. do not enforce it against musicians, only against venues. Whatever it is, I don't see how they could enforce it against all musicians even if they wanted to. Well, I guess they could, but gee there are a bazillion bands playing covers. Far more than those who play only originals. That would be tough to do.