Sahih al-Bukhari » Book of Bathing (Ghusl) » Narrated Mother Aisha RTA
Whenever the Prophet (ﷺ) took a bath after Janaba he started by
washing his hands and then performed ablution like that for the
prayer. After that he would put his fingers in water and move the
roots of his hair with them, and then pour three handfuls of water
over his head and then pour water all over his body.
Sahih al-Bukhari » Book of Bathing (Ghusl) » Narrated Abu Ja`far RTA
Jabir bin `Abdullah said to me, "Your cousin (Hasan bin Muhammad bin
Al-Hanafiya) came to me and asked about the bath of Janaba. I replied,
'The Prophet (ﷺ) uses to take three handfuls of water, pour them on
his head and then pour more water over his body.' Al-Hasan said to me,
'I am a hairy man.' I replied, 'The Prophet (ﷺ) had more hair than
Sahih al-Bukhari » Book of Bathing (Ghusl) » Narrated Maimuna RTA
The Prophet (ﷺ) took the bath of Janaba. (sexual relation or wet
dream). He first cleaned his private parts with his hand, and then
rubbed it (that hand) on the wall (earth) and washed it. Then he
performed ablution like that for the prayer, and after the bath he
washed his feet.
From all the above hadith, it is proved that washing hand in beginning is sunnah, but not a farz that if missed then you Ghusl is not correct.
Overall purpose is to make sure it is properly cleaned, and we all know the 3 Farz which you mention in your question.