Ven. Thanissaro's first point requires that 'the all' of the six sense spheres is an incomplete description of possible experience ("So there's nothing in the Pali discourses to indicate that the Buddha would have agreed with a modern materialist view that experience is limited to the six senses") in order to allow for his understanding that consciousness can exist in a separate dimension beyond 'the all', but this is directly refutable since the Buddha declares that any consciousness is defined according to the concomitant sense activity, the condition for contact - there can be no consciousness that isn't consciousness of
one of the six sense spheres, however one might try to describe such a dimension. (Also, a six-sense epistemology is not at all the same thing as "modern materialism", yet these are inappropriately equated.) Nibbana is a lack of certain functions, not an infinite function in a mystical dimension.
The second point suggests that to not believe in rebirth is to "remain entangled in the questions of inappropriate attention, which will prevent you from actually identifying and abandoning the causes of suffering and achieving the full results of the practice." Yet MN 2 describes how inappropriate attention involves "Was I in the past", "How was I in the past", "Will I be in the future", "How will I be in the future", which seems to be where talk of rebirth invariably ends up. This argument runs afoul of its own criticism.
The third argument involves the idea that "if one's experience of awakening doesn't match the descriptions in the Canon, one would do well to examine one's motivation for wanting to claim a canonical label for that experience." While thinking of the Canon as a homogenous bulwark of authority is par for the course vis-a-vis the Theravada, it's rather naive - even thinking of the four main Nikayas in this way is inaccurate; indeed, differences in which texts are taken as authoritative is reflected in the great jhana debate as well. It's certainly a much more complex textual environment, as a recent post in the Early Buddhism Resources thread sums up.