Haven't read the whole thread but read your posts OP. I see you haven't been back for some days so may not see this but my heart goes out to you and I wanted to say a few things:
(1) It is SO HARD. You are not alone. Anyone who tries to 'perfect parent' at you can sod off - the problem with parenting, especially when they're this small, is there is no let up - you just work, all day every day, either commuting or working or child-rearing or in the home and the only respite is sleep (if you're lucky). Even with easy, charming children and a supportive spouse, it is relentless and you feel like a hamster on a wheel.
(2) having said that, wanting to throw yourself out of the window or 'disappear' - assuming this isn't just hyperbole or a figure of speech - is not normal, is worrying, and suggests as per your UN that you really do need some help.
Please see a GP - don't be afraid of medication. People will give you all sorts of tips to improve your life overall in real terms - I will be one of them below - but in the first instance, you are in a difficult situation, it is not tractable to a simple solution, and you need to be able to cope day to day, to get out of bed and manage and not scream at your little ones or burst into tears on the school run and then have to deal with guilt or embarassment - medication, for me, is a massive massive help with this. Reports vary, but for me a low dose of SSRI daily doesn't deaden me or stop me feeling my feelings, it just gives me a sort of breathing space between my feelings and me so I don't feel so compelled to react to them in a way that will make matters worse. Consider this. People always talk about 'getting help', but realistically there is very little real help on offer, even for people with a proven history of suicidal tendencies in absolute crisis - but SSRIs are to be had more or less for the asking and may be the short term crutch you need to allow you to pull your head above water and get the lie of the land.
(3) I know you said not to fixate on it, but your useless DH is very much the fucking problem and what, as soon as you are able to think all the way to the end of a sentence, you should be focussing on. As I say child-rearing is hard even when you're working as a team; it's fucking impossible if you have someone undermining all your efforts, adding to your workload and shoving their oar in in unhelpful and counterproductive ways. Not to mention your resentment at the millstone that he is around your neck will eat you alive. You say he can't cope, is overwhelmed, shuts down. Are we talking neurodiversity here, or mental health troubles, or just plain old lazy and inadequate? If one of the first two he needs, as a bare minimum, to pursue some diagnosis and treatment as it is 100% not normal for him to be totally unable to properly parent his kids or contribute to family life. And the brass fucking neck of him to ask why 'you can't do it too' when everyone else can - a man who won't even take both of his kids out on his own?
I think if you really look at things, you'll see HE is 100% the problem, not your kids with their perfectly normal boundary-pushing behaviour. Your kids will be confused because he is putting them in control, by giving way to them and bribing them with sweets. You say yourself the kids are different kids at childcare - they will be happy and relaxed there because they know exactly what the boundaries are, that they are firm and upheld, they know what is expected of them and they aren't left in charge - can you imagine how scary it is for a powerless little 3 year old to feel that their own will is stronger than their father's, the person who is supposed to care for and protect them? How can they feel safe, with such a weak parent? They're acting up because that is bloody terrifying for them. They keep pushing because they need to know in their little scared hearts where the boundary IS.
I won't ask how your relationship is apart from around the kids, because once you become parents there really IS no 'apart from the kids' - you are not just a couple now, you are a family, in which everyone has a role to play, and he is opting out. That cannot help but make you resentful of him in every single aspect of your lives. So it doesn't matter if he is romantic, funny, a great shag, a good provider - he is letting you down in a fundamental way that is not excusable. Fundamentally, every other bit of advice people can give you about improving things with your kids as long as he is there gumming up the works and undoing everything you try with his weak-ass 'parenting'. So just sit with that for a bit, and then start tentatively imagining a life without him, where they get his bad influence every other weekend where all he has to do is 'keep them alive', and YOU get every other weekend off duty. Just imagine it. Think of the kind of life you could then build, for you and your boys.
(4) The overwhelm is real, and you NEED a break. You say you have no family in the UK; can you go, by yourself, for a long weekend, to a supportive friend or family member abroad? Or are we talking transcontinental? Or failing that, are there funds available to just check yourself into a hotel/AirBNB somewhere green and just rest, 100% rest, alone, for a couple of days? Useless husband can have a weekend away in the kitty for it if you must (although I'd be willing to bet money he gets out on his own lots already). But right now you need to break the circuit. Prioritise this over almost anything else. Reset. You have all the right ideas for improving things with the boys - focussed attention, better boundaries, getting out in green spaces - but right now you are so ground down any of that seems impossible to enact. Once you've had some restorative time alone, it will seem much easier.
(5) Cleaner, and laundry service. You work full time and have two kids. You have enough to do. You don't need to be doing daily chores on top that can be farmed out elsewhere. Alternatively, if DH 'can't cope with' and is 'overwhelmed by' his own kids (like you're not?), he has to step the fuck up on everything else - housework, laundry, cooking - while you focus on the behaviour and on getting some downtime to enable you to cope with that. He does not just get to cruise along pleading 'overwhelm' while you work yourself into the ground.
TL;DR - ditch the useless husband, give your kids the boundaries, focus and nutrition they need without him around to undermine you, and prioritise your own mental health as a matter of urgency. You have this OP.