Part of your job as parent is to be raged against by grumpy children sometimes. You're the safe people they can do this with when they're frustrated about something.
Not being able to keep everything due to space is frustrating, but it's a lesson they need to learn before they turn into hoarding adults keeping everything 'just in case'.
As parent I think you shouldn't try to avoid being the one faced with their frustration by treating them like mini adults and expecting them to make these decisions all by themselves. Yes they're separate people with their own feelings and thoughts but they're not adults and shouldn't be treated like they are.
As lots of pp have mentioned it's unlikely they'll even notice if you collect up and throw out all the broken things when they're not there. If you collect up all the things you notice they don't play with, keep them for a couple months, if they don't ask where it is in that timescale they're unlikely to ever ask, so donate these things. If that doesn't create enough space, give them each a box and ask them to put all their favourite things in there.
Don't ask them what do they want to give away, they're children and they don't want to give away anything. Ask them instead what they most want to keep, tell them it all has to fit into the box.
Then collect up the rest, keep for a couple of months in case they ask for something back and then donate. Do the collecting up and donating when they're not there. If they ask where the stuff is much later, remind them they got to choose and keep their favourites, explain there wasn't the room to keep the rest. If they're annoyed about it, suck it up. They've a right to their emotions, they're children who don't understand about mortgages and why there's no space. If they can't express their emotions to their parents then who can they express them too?
At less than 10yrs old they don't deserve to have to make what to them will feel like a major life decision about which toys to part with. They're too young to shoulder all the responsibility for their actions. So let them choose their box of favourites, then if there's any regrets you take the flack for being the 'bad guy' who made them get rid of their toys.