Middle aged family man here. The way I think most "handle it" is by having less on their plate and balancing work and life in a different way than you're describing.

1. For me the most important factor is partnership. My wife and I split up our responsibilities equitably and we each play our roles well. We're also flexible enough to cover and support each other when needed. If you can't do that for each other you don't have a partnership.
2. The second most important thing is a strong support system. We intentionally moved to a place where we have a lot of family close by before starting our own family. My inlaws make it possible for my wife and I to each work a 40 to 50 hour week while ensuring our kids have a rich home life and don't miss out on anything. That doesn't have to be family though. It could be a mix of school and after care, or a church, or friends, but if you don't have some support system you will eventually collapse under the weight.
3. Pick your battles. It's OK to have takeout or heat up a frozen dinner if you don't have the bandwidth to cook sometimes. My house is always clean and sanitary, but it's also constantly messy.
4. Like many things in life there is an element of attitude to it. If you give in to defeatism it's easy to spiral. If you view your family or home life as a weight on your shoulders you're doomed. That should be the wind at your back. That should be the stuff that lifts you up. That's entirely on you to sort out. IMO you should probably talk to someone about it.

Overall What your describing is that you've built your life in a way that doesn't work for you. And to your point a lot of men who do that opt for solutions that make it worse (affairs, substance abuse, etc...). You're not going to wake up tomorrow and things are suddenly better. At the very least , you need to take active steps to find a better job or work out a different balance of responsibilites with your partner.

Good luck, stay strong, I'm rooting for you.