I have not read Built to Move, but I have read a couple of Tim Anderson's books, and spent a significant portion of my own time learning about "movement." Just wanted to drop two cents:
I would look for books that are more princple-based than "drill- based." Everyone seems to have a slightly different interpretation of what "movement" and/or "mobilty" means. I have Move your DNA and enjoyed it. Have not read much Starrett.
For general health, the two biggest things you can learn about are the gait cycle and breathing. These are the two kinds of movements you do the most. Breathing is a given, and although many of us ought to walk more, sometimes (not always!) the way we walk and stand upright affects the way we move in general.
So learning some PRI (postural restoration institute) principles would be helpful. I am not a PRI fanatic; I think some of it's online proponents get a bit off in the weeds, but the core ideas are great. Conor Harris' youtube channel/blog is a good start. A lot of what he uses and demonstrates comes from PRI.
The book that had the biggest impact on my approach to movement in general is What the Foot? by Gary Ward. Life changing. While he does focus on the foot, it is his approach to how muscles and joints behave in systems that really changed things for me. You might benefit from some basic skeletal and muscular anatomy knowledge to understand certain parts. He's not suuuper technical, but you might get a bit lost if you don't know bones and muscles that well.
He is heavily focused on gait, and as such, some of the principles he puts forth need reframing when we are talking about training with weights or something. But for general movement overall, his ideas are gold, imo.
If you are interested in such things, I also suggest reading from as many different sources as you can, and then finding what common ground(s) they share. You might be surprised to see the diversity of approaches and ideas out there. Then, always see what approaches work for you.
You can never really go wrong learning basic anatomy, and then applying that knowledge to other peoples' movement approaches, and seeing what works for you. Learning joint and muscle actions is important, but it's equally important to remember that a great deal of anatomy books use "dead guy anatomy." That is, anatomy of a person being dissected on a table, NOT a person in motion. The actions/roles of certain muscles change based on position and the action of muscles which also attach to the same joint.
Thus, my suggestion for a broad approach, depending on how serious you are or how deep you want to go.
