Mine has always loved books from when he was tiny, fabric books, board books, flip the flap, number books, rhyming books - everything - but it doesn't mean he always sits still or always looks like he is listening. But he IS listening. If I stop reading he notices and tells me to read, if I stop and ask a question he knows exactly where we are up to in the book and can answer the questions. He will and does sit still for books and likes to snuggle up for them on the sofa, but equally he will play, roam, fiddle or watch the TV at the same time - and yes he can follow both the TV programme and the book and know exactly what is going on with both! We don't do (and never have done) BT books, instead we have a bath time book or two and heaps of books at any time of day, in the bath he plays, chats, sings and still knows where we are up to in the book. It can be off putting for the reader but it doesn't bother me and I've taught DP not to mind it too (he did feel like DS should look and listen). Other times he is in the bath he pays 100% attention to the book and stays quiet throughout. I had to limit books when he started using the potty as we'd be stuck on the potty for up to 15 books at a time! Madness!
I think if a LO is asking for books or appears to like them being read even if they look like they aren't listening it is well worth the time to continue to read. We don't yet know what sort of learner they will be. Kinaesthetic learners take information in through movement and if they are forced to sit still (which they will have to do through most of their schooling) they can lose interest as the experience lacks the same level of excitement and enjoyment for them.
I feel lucky that DS has never damaged books, I wouldn't like that, I think you can stop them if it is important to you, in the same way that we can, do and should stop them putting their hands on a hot oven. The oven obviously is dangerous whilst damaging books isn't but if it is important to you then the same teaching methods can be used to great effect. There's only been a couple of times I took a book from DS and told him I need to protect the book, and have offered him newspaper or junk mail instead and told him he can tear that up (as tearing and crumpling are important skills, fine motor development). He chose not to tear it but the message was given yk.
He was given 'that's not my baby' when he was born and has always loved it, the mirror at the back is a real pay off to be anticipated.
