Those who extended the wiping up to the armpits did so because one of the chains of the ḥadīth of ʿAmmār contains the wording:
“We performed tayammum with the Messenger of Allah—peace and blessings be upon him—and we wiped our faces and our hands up to the shoulders.”
Whoever treats the reports that mention the elbows or shoulders as merely recommended, while taking the ḥadīth of ʿAmmār as establishing the obligation, is following a sound method; for reconciling texts is preferable to setting one aside, according to the scholars of legal dialectics. Such reconciliation, however, should only be adopted if those other ḥadīths are genuinely authenticated.
Second Issue
Scholars have disagreed about the number of strikes on the ground required for tayammum. Some said one, others two. Among those who said two, the majority—by which I mean the three jurists Mālik, al-Shāfiʿī, and Abū Ḥanīfah—held that one strike is for the face and another for the hands. A smaller group maintained that each of the two areas requires two strikes: two for the hands and two for the face.
The cause of this disagreement is threefold: the Qur’ānic verse is terse on the matter, the ḥadīths are at variance, and analogising tayammum to ablution in every detail is not universally accepted. The sound ḥadīth of ʿAmmār establishes only a single strike for both the face and the two palms together, yet other ḥadīths mention two strikes. The majority preferred those latter reports because they analogised tayammum to wuḍūʾ.
Third Issue
Al-Shāfiʿī differed with Mālik, Abū Ḥanīfah, and others over whether it is obligatory to transfer dust to the limbs during tayammum. Abū Ḥanīfah and Mālik did not regard that as mandatory, whereas al-Shāfiʿī did.
The root of their disagreement lies in the preposition min in the Almighty’s words, “...then wipe your faces and your hands with it” (Q 5:6). Min can denote “a part of” something, but it can also serve to “distinguish a genus.” Whoever understood it here as partitive made transferring the dust obligatory; whoever took it as distinguishing the genus said the transfer is not required. Al-Shāfiʿī preferred the partitive reading by drawing an analogy between tayammum and ablution. This view, however, is countered by the aforementioned ḥadīth of ʿAmmār, which states, “then blow on them,” and reports that the Messenger of Allah—peace and blessings be upon him—performed tayammum on a wall.
You should also know that the dispute over the necessity of maintaining sequence and immediacy in tayammum is identical to their dispute over those same issues in ablution; the causes of disagreement there are the very causes here, so there is no need to repeat them.