I'm a Year Manager (non teaching - I'm not a teacher!) and I work alongside a teaching HOY. As mentioned above, I do all the pastoral/attendance stuff for the year I work with (currently 8 but we remain with our year until they leave), as well as a lot of the data crunching/paperwork. My HOY is bloody amazing and we make a good team. She has a pretty full teaching load, so whilst the buck nominally stops with her, we're a team. Every year has a HOY and a YM, years 12 & 13 have one who oversees the 6th form.
I don't think there are particular teaching/non-teaching divisions. I'm hugely aware that I have no experience/knowledge within a classroom setting, other than what I've seen sitting in on lessons when I'm observing a student within my year (NOT observing the teacher!) and I have no desire to be a teacher. Certainly the relationship between HOYs and YMs is good, as is our relationship with SLT. I have a good relationship with all the teachers (as far as I know) and I am very aware that when I leave the building, I basically leave work. I have huge respect for the teaching staff -I'm wrecked by the time I go home and I couldn't mark if my life depended on it.
I can't comment on whether this system means that teachers kick stuff upstairs as I've only ever worked within our structure. Year Office stuff is used when the usual systems (HoD etc) have been exhausted and whilst there are a few staff who want to hand everything over to the YO straight away, we manage this.
It also means, as previously mentioned, that I'm on call to address issues when they arise and means that when the HOY finishes in class, she gets the information in her hand and can then make decisions based on this, rather than having to charge around school speaking to students. It also means that parents can call the year office when it suits them and means that we can forge good relationships with parents - even if they want to speak to the HOY I can usually get the gist of what the problem is so the HOY returns that call a bit prepared, rather than just returning a voicemail message from a parent asking her to call them back.
I also believe that it can help get on top of low level stuff with the students. They're reasonably happy to come and let me know what's going on so, for instance, we can nip the bullying in the bud in the early days.
I could write a book about how much I love my job and how I can't imagine doing anything else. However, I would like to be clear that I don't earn anything approaching a teaching salary, so I get all the nice bits, but equally, I don't get paid for the tough stuff