Abu Bakr al-Humaydi (may Allah have mercy on him) said: I heard Yahya bin Sa'id al-Qattan (may Allah have mercy on him) ask Sufyan bin Uyaynah (may Allah have mercy on him) about this hadith, namely: "We performed tayammum with the Messenger of Allah (peace and blessings be upon him) up to the shoulders." Sufyan (may Allah have mercy on him) said: I was present with Isma'il bin Umayyah (may Allah have mercy on him) when he came to al-Zuhri (may Allah have mercy on him) and said: O Abu Bakr! People object to two of your hadiths! He asked: Which are they? He said: "They performed tayammum up to the shoulders."
Al-Zuhri (may Allah have mercy on him) said: It is narrated from Ammar (may Allah be pleased with him): "We performed tayammum with the Prophet (peace and blessings be upon him) up to the shoulders." Isma'il (may Allah have mercy on him) said: The hadith of Ubaydullah mentions touching the armpits.
It is narrated from Ubaydullah that Sayyiduna Umar (may Allah be pleased with him) ordered a man to perform ablution because he had touched his armpits.
The hadith about touching the armpit is mursal: Ubaydullah bin Abdullah bin Utbah did not meet Umar bin al-Khattab, and al-Zuhri disowned it after having narrated it.
Sometimes the order to wash the hands is given for the sake of cleanliness. And Allah knows best.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 655