This is clearly an emotional issue more than a financial one. Your car is demonstrably safe -- safer now in one respect, as it will soon have a brand-spanky-new throttle body or cable. And yet given what you say about your losses this year, I am not at all surprised that your wife is drawing the line and saying "I won't lose you too."*
Given the emotional and relationship issues, I am not going to encourage you to hang on to what is clearly a perfectly safe and reliable car (at least considering its age). Your wife is dealing with a lot of shit, so this is probably not the hill you want to die on.
At the same time, you don't want to get into a habit of fixing emotional problems by throwing money at consumer goods. Have you and your wife seen a counselor or therapist to deal with the losses? Do you have a support network that can help with practical things to get you through (food/kid pickups/etc.) or just to give you an opportunity to go out and blow off some steam? That would be step one for me. Everyone needs a little help some time, so make sure you guys are getting what you need to deal with the actual issues. Otherwise, you are just throwing your money away on a symptom without dealing with the real cause.
If you are getting the help you need and are just in that period where you're working on it but it's not better yet, then I think it's ok to add a little pure fun to your life. I think this is one of the things that tends to go away when you are dealing with tough shit, because you get so bogged down in dealing with all of the extra stuff along with everyday life, and you don't have the energy to go out and just do something frivolous. But the first step here is making time for fun things that don't cost $20K!! Rent a ridiculous movie (Monty Python, Mel Brooks, or whatever floats your particular boat). Buy Jiffy Pop, just because it's awesome fun to watch the little metal dome unfurl. Go to a park and play frisbee or fly a kite, especially if you're bad at it. Play hooky and go see a matinee. Etc.
Now, car: I am a car girl. I love driving. It is perfectly reasonable to want a car that is fun to drive. But you don't need to spend $20K to do it. If you like the Focus ST, what about an older model? There are a bunch of hot hatchbacks and the like that have been in production for a decade or more; I'd bet if you are patient and look hard, you can find one for a reasonable price.
In the meantime, if you're a little short of what you need to get the make/model/year you want, why don't you just sock away your extra cash towards a car fund? Make a deal with your wife: you will trade in the Chevy and get something sportier, but you're going to pay cash for it, and towards that end you will divert $X of your $Y monthy savings into a car fund until that fund is sufficient to cover the (older/used/still peppy) car. In the interim, you have a cheap car that is still perfectly drivable -- and you can work on the therapy and free fun, and maybe buy yourself a few months to see if it's really about the car or if the car is just a substitute for something else that's missing.
*I also dumped a car after it blew up on me in a parking lot -- yeah, ok, it was just the radiator expansion tank, but it sounded like a bomb went off and dented the hood, and I realized that if that had happened on I-95, the kids and I could both be dead. I justified it because the guy at the shop told me about all the other stuff that was going to require thousands more dollars within the next year. But, really, there was just no fucking way I was putting my kids back in that car. Totally emotional decision.