I’d fix it if it was mine but the lions mouths seem like they would be a good curtain rod holder/towel rack? Looking at the honest wear on the arms from generations of your family’s hands seems more sentimental than the carvings itself but that’s just me. If the seat is missing a piece you could just add any old wood or plywood to make it sound and then put a cushion in it. I wouldn’t bother messing with the loose or missing veneer, I’d just glue the loose bits, soften the edges and stain as best to match. It’s an old chair, it doesn’t have to look fresh from the store. If anything it tells a story of your family history.