If you want to save your marriage you are going to have to do something about that.
You are going to have to delegate some responsibilities, and you are going to have to get yourself to bed earlier. May not be easy, but bottom line is you're going to have to do it. The way your husband sees it, HE works all day too...and HE's not too tired. So just as he says, there's no excuse for you to be either. Sex is not a marathon run. Getting "strenuous" with it is optional, not mandatory. There's no reason just being "tired" should make it impossible for you to do so much as lie back and enjoy yourself - and let him know you appreciate being close to him in this way - even if he does most of the "work." When it comes to sex, in the eyes of husband, "tired" simply doesn't cut it as a reasonable excuse for a wife to make, and I don't care if she's got 15 kids.
Now, how do you prove it to him? You talk to him...outside of the context of sex. You perhaps "suggest" some of the points I've already made, and ask him to confirm it. "Have I rejected you so many times that you've given up?" "I'm sorry - REALLY I'm sorry." You've got to get him to see that you know you've hurt him, and that you really do miss him (yes, you honestly do, because you wrote this thread about trying to reinstitute sex in your relationship). And you know it's going to be hard for him to believe that...and that you're going to have to show him.
To just rush up and initiate sex would pressure him. He wouldn't believe you, he's likely to just push it away. You've got to show him that you really care about how he feels, that you understand better now how sex really is important to him, important enough that the way you shrugged it off as something you were "too tired" for was inexcusable and you know it and you're not going to do that again.
And if there are specific things that you can tell him that would help you not be so tired....remember I said SPECIFIC...then tell him. The more specific, the better. Men think concretely, men think like accountants. You have to show him that he's going to "get his money's worth." Sorry if this sounds sexist - I know it does - but it's the truth. If his taking the kids for a Saturday afternoon would give you some time to recoup and you could promise him sex on Saturday night....then do so. And it's perfectly OK and justifiable to tell him that if he wants you to not be "too tired," he's going to have to do more of the housework even when he gets home from work, just like you already do. You can promise him sex if he does _____ (insert chore here) before you go to bed. BUT WHATEVER YOU DO, DON'T BREAK THAT PROMISE. Even if at bedtime you really DO get a headache, you take an Excedrin and you grit your teeth and you go through with it. No, this doesn't have to be a permanent condition of the rest of your life, but certainly while you're rebuilding his trust. You're going to have to keep your word, get him to trust you again, that when you say you're going to be there for him in bed, you really mean it and you're going to follow through on that promise. "If you do the dishes Tuesday, we'll throw in a quickie then too." "Arrange something for the boys to do on Sunday that gets them out of the house, and we'll have a wonderful time here all by ourselves." You'd be amazed what a husband might do with motivation like that. Most of us truly will work for sex. But you have to do your part and make it worth his while, and that means consistently and reliably holding up your end of the bargain. Disappoint him too many times, and you'll be right back where you started.
It took a long time to get down to where you are now, it's going to take a while to get back up to where you want to be. But be consistent and be trustable, and you will.