The scope can be as narrow as "do background checks on hires, do the math on their desired pay and tell me how much each one would cost and if there's any red flags" or as broad as "do a first phone call and check out if you think that person is a good fit culturally." Handle all associated paperwork, set up interview dates, handle travel and accommodation arrangements for the prospective hire, ... Handle all paperwork associated with employees, sick leave, keep track of holidays, ... Follow changes in applicable law. Organise team building events. I know of an HR department that organises monthly lunches where people of the company get randomly assigned to groups so that they get to know other people working in the same company, not only their team peers. There's a lot of things you can do as HR person. Not everybody needs everything and not every HR person may have the required qualifications for each task, so you probably just have to figure out what exact set of responsibilities right for your place.
Out of curiosity, could you imagine an organization functioning where HR work was done by administrative assistants to team managers?
No. Mission creep happens on broad and narrow roles the same. The better defined the role, the better. Constant reevaluation and adaption to changed realities is a must.
> Out of curiosity, could you imagine an organization functioning where HR work was done by administrative assistants to team managers?
Up to a certain degree yes, but there's the question how and where to allocate common functions if you have any. Depends on your organisation. That doesn't make the "HR department" disappear though, it just changes its name to "administrative assistants". It certainly makes things harder when people try to reach the HR department because that's what they're used to.
All in all: If that fits your organisation, sure. Just make sure that the role is defined and someone is responsible and actually takes up the work.
edit: see also https://news.ycombinator.com/item?id=6289282 for a point where a more centralised HR made a positive impact. An assistant to the team manager might easily have missed the broad picture because of a more narrow view, focused on the teams turnover rate.
If you get a moment, read "Corporate Confidential" by Cynthia Shapiro. Only 200 or so pages, and it's a quick read. But very enlightening.