What do you do if a piece doesn't take off? I was thinking with the method book that often a child finds one piece hard or it just doesn't appeal to them but they take to the next piece. How insistent are you that they can play the earlier piece correctly before they go to the next one ?It's a good question. Not as insistent as I used to be. I don't think "getting bogged down" teaches anybody much at all, except perhaps that music is boring

and I've gradually got much less tough on finishing every piece. I find generally that the piece that doesn't take off in the first week or two rarely does so later. These days I tend to explore pieces briefly to give pupils plenty of choice, and I try to work on the ones that seem to appeal.
I'm not happy about leaving things unfinished on the whole - but if I feel we have got all we are going to get out of it, I usually find a reason to move on. If the pupil can get through it in a fairly basic way, I feel has made some effort, and is clearly not enjoying it, we usually agree to give it a "small tick" and leave it.
If it pertains to some important point in the tutor book I make a note that I need to find something else and approach that point in perhaps a different way another time. And of course if the next couple of pages in the book build on that piece you probably need to find other things to reinforce before moving on. If most pieces are ending up as "barely finished" though I need to make the point eventually and perhaps be a bit less lenient.
An idea I picked up from Philip Johnston's Practice Spot website was to award an extra credit for the pieces that are particularly well completed. (Yes we use stickers too but they don't seem to act as much of a carrot, although they are regarded by most as quite nice to get). The idea is you collect the credits and when you have so many your name goes on the credit chart. (You decide what to call them and how many are needed) With these I find that pupils tend to keep playing the pieces they like long after they get a tick for them, and then will sometimes ask to play them again a few weeks later - and surprise me. I have found this very useful with a number of younger pupils.
Don't know if any of this helps. It does vary a lot from pupil to pupil - some will "go off" almost every piece they start given half a chance! Others don't like leaving a piece half-finished, and seem to take a while to get into a new piece but get there in the end. I don't claim to have all the answers - or even a few percent of them.
By the way I don't find that piece books are regarded as different from course books - or I haven't picked that up. I do like pupils to have more than one thing on the go at any one time, if possible pieces at varying stages of completion, and preferably from different books.