You know, to me you sound a little defensive, but I think it's unwarranted.
You're trying to do so much, your kids sound essentially happy, they know how much they are loved and have joy and security around them 24/7.
If you don't take time to recharge (be it at the gym or having a nap) how could you possibly give them as much time and energy as you do when you're home? - and lets face it, if my children weren't sleeping through and I could I would be taking a nap too, (and have been known to do just that if dp is home and the oportunity arises) and I would probably be a better mother for it.
I think you're doing a great job, don't beat yourself up for some things so that you can do the rest of it that much better.
Your kids don't need a short tempered zombie who is doing too much to even be able to enjoy them, they need a mummy and that's what you are giving them