First of all, my heart goes out to you. Homeschooling can be very frustrating...expecially with pre teens.
We all can only speak of our own experiences and I don't know how that can really help you. You have a unique and individual child...just like the rest of us. Each child responds differently to teaching methods and learning in general. If what you are doing isn't working, shake it up. Change curricula, enroll him in one class somewhere and see how he responds to outside instruction. Some kids desperately want mom to play the mom roll, the support system, the care giver and disciplinarian....but perhaps not the teacher? I can't vouch for what sort of education options you have in your area but you typically don't have to do it all yourself. We have outsources a few subjects and have learned that our son (at age 12, now 13) really thrived with some outside instruction. He felt a little bit of independence, he learned he could really do it on his own...it gave him pride in his efforts and he was thrilled to have mom and dad see that other people liked what he had to say/write/do. At this age, I think many children need this.
I don't know if enrolling him in full time public school will fix the problems. There are many issues to consider and only you know your child well enough to weigh the balance of costs and benefit. I suppose this letter isn't really of much help but....listen to your heart, and your child - he is old enough to try to figure out from where his apathy comes. He is also old enough to reckon with issues of disobedience, sin, sloth, deception and a myriad of other issues. We can't begin to know the heart of those issues...but I bet together you can figure them out.
Good luck whatever you choose...and many blessings to you~
Janine