I certainly wouldn’t run to HR right away — but unfortunately, it’s true sometimes that people just aren’t a good fit for whatever reason. Deadweight that isn’t able to accomplish the tasks that need to be done doesn’t do you any favours — if you’re doing your job and their jobs because they just can’t handle the tasks that’s hardly fair to you, and isn’t doing the organization any good — eventually you’ll burn out, nobody will pickup the slack, and everyone will suffer for it.

My first instinct in your situation however would be that everyone has got used to the status quo, including the staff you have to constantly mentor. Hopefully if you can coach them into doing the work for themselves and keeping them accountable to tasks and completion dates will help change the dynamic.