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Maybe I'm missing something, but that looks like it should be built of brick (whatever the correct sort is considering water penetrating,weight etc.). Then you cap the steps in a paving stone of your choice.
So yeah, smash it out and build a new one out of brick, render/paint, pave.
Given there could be supporting elements in relation to the wall I'd probably pay someone. But then it might be one of those jobs no one wants to do because they can make more and easier money building rear extensions.
Are you within @ColinTheBald 's geographical reach? Looks like the sort of thing he could do along with paving the bottom.
If you've got young kids something you should really consider is whether a slide can be added to the left at the same time. It would be such a cool feature and make them less likely to fuck themselves running down the stairs.
Not to hand, though here's a screenshot of a photo from when we moved in and, apart from scrubbing and a lick of paint it's not improved much.
The side return is made of 3 tiers/terraces, one underneath whatever that blue thing was (which includes a manhole cover leading to the sewer about 20 ft below), one where the photo was taken from, one behind. All about 20cm step each.
Wall to the right borders neighbour. Crumbling at the top with overgrown shrubbery. The wall itself is fairly sound (chonky Devon stone) but the top is angled crumbly concrete and a strong wind would probably see it gone.
Width of current stairs is around 90cm from memory.
Stairs currently around 3m long I think. And return is about 10m including stairs. Don't have drop measurements to hand.