As for the hadith of Amr ibn Hazm, it is weak because it is mursal—that is, its chain is missing the Companion—and a mursal report is one category of weak hadith. A weak hadith cannot be used as evidence to establish legal rulings, much less to establish a ruling that imposes severe hardship upon Muslims by requiring Allah’s servants not to read His Book unless they are in a state of purity, especially during cold weather.
Even if we assume it to be authentic on account of its widespread recognition, the word “pure” may mean one whose heart is free of shirk, one whose body is free of filth, one who is free of minor ritual impurity, or one who is free of major ritual impurity. These are four possibilities. When evidence admits of two possibilities, it can no longer serve as proof; so what then when it admits of four?
Likewise, the term “pure” may be applied to the believer, as Allah Most High says: “Indeed, the polytheists are impure.” [al-Tawbah: 28] This affirms impurity of the polytheist. The Prophet ﷺ said: “Indeed, the believer does not become impure.” (1) This negates impurity from the believer, and negating one of two opposites entails affirming the other, for there is only purity or impurity. Thus, it does not indicate that whoever touches the muṣḥaf must necessarily be in a state of ablution.
As for rational consideration, we do not accept analogy at all, for the Zahirites do not affirm it.
In my view, their refutation of the argument based on the verse is clear, and I agree with them on that.