I know the discussion is on Paul Washer in particular and the nature of assurance in general.
It occurs to me that a conversation I heard recently between Sinclair Ferguson on the Theology on the Go podcast points (in part) to one of the issues many Reformed people have in discussing any theological topic. He was pointing out how Hugh Ross' writings serve to remind us of the mooring that our theology has in the Covenant of Grace. I often think many discussions about faith or assurance tend to leave their mooring in the Covenant of Grace and are taken up independently as if they can be considered without understanding how Christ's Mediatorial Offices operate to secure and grant these Evangelical graces to believers. Conversations between the neophyte or the nomist or the antinomian with the preacher of the The Marrow of Modern Divinity come to mind.
We tend to discuss the issue of works, for instance, in the abstract, quite apart from their connection to whether a person is in the Covenant of Works or in the Covenant of Grace. It is the same thing with the Law itself.
Once we move apart from the ground of the Covenant and start to discuss faith or works or assurance then we are divorcing them from the whole Christ and how He works them together in the believer.
This is why a person operating from a lack of Covenantal understanding can read some Puritans (or their followers) and see only in them a morbid introspection.
One can say that assurance is of the essence of faith because it is the instrument by which we lay hold of the whole Christ in Whom assurance is found. It is a sinner, however, united to the Whole Christ and while sanctification is also of that essence of being united to Christ, it moves in fits and starts. We see the flesh still warring within our members and we are called repeatedly by the Scriptures to reckon who we are in this Christ as His Mediatorial offices of Prophet, Priest, and King are being outworked visibly and invisibly in the lives of Saints.
I can't speak about Washer, but I can say that one cannot judge a particular "reaction" to any sermon based on the hearer who is operating in a way that chops up theological bits like faith, assurance, good works, sanctification, etc as if they are to be tossed about in the mind as propositions to be logically understood without reference to the fact that they are all within the outworkings of the Mediator.
Incidentally, though not related to assurance, per se, this is the issue that is roiling within my own denomination and "Side B" Christianity. It's yet another example of sinners not grasping what it means that they have been brought from death (in Adam) to life (in Christ). It changes the entire conversation. It changes the whole outlook on what the Law, works, holiness, etc looks like. When one loses the plot (Covenant) then it affects everything and people even forget that sinners are united to the whole Christ and what that really means for the way we talk about ourselves to include our doubts, our sins, our works, etc.