Miranda (from the case Miranda v. Arizona) is protecting the fifth amendment protection from self incrimination (“You have the right to remain silent”). The right to “assistance of counsel” is in the sixth amendment, and Gideon v. Wainwright established that that means the state must provide public defenders to the indigent.

Another fun fact, the Gideon case only provided for public defenders in felony cases, and a later case, Argersinger v. Hamlin, extended this to any case that could result in actual jail time.