For a second example, consider Martin Heidegger. Somewhere in Sein und Zeit he writes that Das Dasein ist nie vorhanden. The human being is never present-at-hand. This is obviously false in that the human being has a body which is present-at-hand in nature as surely as any animal or stone. What he is driving at is the truth -- or at least the plausibility -- that the human being enjoys a special mode of Being, Existenz, that is radically unlike the Vorhandenheit of the mere thing in nature and the Zuhandenheit of the tool. So why doesn’t he speak the truth, and nothing but the truth, without exaggerating?
First, we are told somewhere in BT, not very reasonable for a philosopher, revealing already the beginnings of uncharitability. Next, without really arguing for his disagreement, Maverick Phil makes an appeal to something "obviously false." That's not really an argument in itself, but simply an assertion. As any interlocutor knows, a the disagreer has the burden to offer a replacement view to advance the intuition of falsity of the target position in question. You can't just blurb out disagreement and call it an exaggeration.
Now, there is an analogue to Heidegger's intention. Well, sorta. Consider the Third Antinomy of Kant in the CPR. Being a mere thing in nature understood from the third-personal viewpoint of science (the impartial viewpoint, the view from nowhere) is a particular way of being and can be found historically in Kant's Third Antinomy. Most of metaphysics and philosophy has assumed this is a primary mode of being, and place human beings in this type of understanding/framework/interpretation. Upon phenomenological reflection (from the first-person point of view, the phenomenological attitude), however, we experience ourselves as acting under the conception of freedom, what Kant calls transcendental freedom. You can understand Heidegger's intuition in the same way Kant will privilege a practical mode of being, the mode of transcendental reflection over an undrestanding that places human beings as simply an object to be subsumed under causal determinism of the third-person viewpoint. In this way, so does Heidegger. He finds that we are in a primary mode of being as absorbed in the practical affairs of our everyday life and we can consider this a more narrow conception of the first-personal viewpoint. Moreover, this is primary whereas the viewpoint from nowhere is an unrealistic abstraction. We can make sense of Heidegger's motivations in a consistent way that doesn't mean he is exaggerating anymore than the assertion of him being wrong.