Because staff is a collective noun referring to a group of people and is therefore already plural. The sentence works with the members as 'newly recruited staff'. You can however work with several staffs if you are working with entire groups from different companies. The plural staves you refer to only applies to to a different kind of staff which is like a large stick. If you have several large sticks then they are staves.

As you pointed out in the title, "staffs" doesn't exist: it's either "staff" or "staves". The staff/staves designate the entire "crew" (group of people/the employees), that's why you need to add "member(s)" to refer to some specific people within that group.

Staff is an uncountable noun, so it can't take an 's'. Think of 'staff' as having the same meaning as 'team'. If there are 11 players in a football team, these 11 people are not all 'teams', are they? They are 'members of the team' or 'team members'. And the word 'staff' is just the same. If there are 20 people working in a company, together these people make up the 'staff'. And - just like with the football team - these 20 people are 'members of staff'. Take my advice - forget about the word 'staffs' and 'staves'. They might be in your dictionary, but all they have done is confuse you. They are very unusual words - I have never said or written either of them in my whole life. Sometimes dictionaries cause problems rather than solve them.