Is the figure of independent counsel so powerful in the justice system?

Yes, but it is less powerful than it was in the Nixon years, where it had near abssolute power to investigate. Back then, the SP had a lot of independence – he could still be fired (Read up on Archibald Cox) but it was a bigger deal.

Now the position is much more constrained and politically motivated.

The problem is that indep. counsels are by definition an aberration – an exception to the standard administration of justice. And therefore no one knows what to do with them and how much power they have.