That first sentence is debatable. One might argue that if you support the principle of Private Interpretation of Scripture, you also support, condone, make possible, all those movements that push that method to extreme, often contradictory conclusions. You can argue “I don’t agree with those who don’t follow Scripture rightly” but they could say the same about your position. Your defense of P. I. makes possible (validates) the snake handlers in Tennessee and the recent Presbyterian conference.
By the same token, there are those who support the principle of Private Interpretation of the Scripture/Tradition unity. For many years, this seemed more secure, less prone to wild variations away from doctrinal orthodoxy than Private Interpretation of Scripture alone. C. S. Lewis regarded Anglicanism as an imperfect system with self-balancing mechanisms that succeeded in the long run to maintain orthodoxy. Anglicanism had internal authority, or rather authorities, but not the single, worldwide Magisterium authority that another group has, so all Anglicans are “private” in that sense. You choose your own authorities. Every vile church position since 1965 has been thoroughly wrapped in Anglican and other historic Christian quotes, even if it really came from the NY Times.
In Lewis’ lifetime being “private” (i. e., non-papal) wasn’t a big problem, as almost everyone in my second paragraph more or less followed orthodox doctrine anyway. Today, Anglicans in ACNA or the Continuum disagree with Private Interpretations taken by other Anglican bodies to positions on Ordination, and other things. But if you assert the right to P. I. for Scripture/Tradition, then in effect you unwittingly justify the right of other Anglicans to follow it to those other places. And maybe worse to come.