'Do you find your children are happy to sit down and learn/study what you've got planned?'
Not always! Like everyone there are certain sections of knowledge my children would rather leave out and other things (history and writing for DS1, science and drawing for DS2 that they could do all day) But my personal view is that there are some things - mostly the basics or core knowledge that has to be covered.
When I first started home ed, my children out and out refused to do any sit down formal 'work'. Initially this was because I tried to take a sort of 'Teacher' role i.e., telling them what to do. I realised my mistake and now find that my children work best with me working with and alongside them. It is perfectly possible to plan a formal set activity from a book or unit study and yet work through the material together (and in my experience you nearly always find out something new yourself) co-operately and make it fun and interesting without it being 'dry'. I strive to explain the relevance of anything specific I wish them to learn beforehand so they know I'm not just trying to push things which have no purpose in their lives. Planning some other hands-on game or art work or other activity or craft related to the subject helps to make what they're doing fun. Breaking it up every so happens helps - We often end up going off tangent and exploring other things. We do lapbooks projects, as well as plenty of reading and nature observation.
There are also bad days when I'm not so involved, busy, tired, and nothing gets done (and i'm tearing out my hair), or they would rather do what they want all day. Personally I find that as we home ed they have much more time than other children to pursue their own interests, and loads of outings and outdoor time, so I don't think I am being unreasonable to expect them to do what I ask for a couple of hours.
Overall to be honest once the routine is established they gradually become more independent and don't wait to be asked to get their books out etc. My Aunt who is a teacher came round the other day and was amazed to see DS12 working through his math book completely independently and DS9 checking his spelling with the dictionary whilst we chatted in the kitchen and made tea. If you have certain expectations they will reach them I think.