Just some more thoughts on why you should go for the flat. We used to live in a similar mid-century estate. We moved when we had DC2. What would have enabled us to stay was if our flat was bigger (it was a tiny 2-bed, approx 600 sq ft) and if we had better access to the communal outdoor space (it was down the corridor, several flights of stairs, several doors and a gate that the caretakers often forgot to leave unlocked on weekends - so use was limited).
In your case, neither of those things would be an issue. The bedrooms are big enough that DC can share until they're quite old (potentially forever if they're the same sex). There's loads of storage too, which is a big plus in terms of making flats liveable with kids. The direct access to outside space that everyone else says is a negative, I think is a positive. You can see from the pics that the current owners use the area directly outside the door as an almost private space. The tiny third bedroom has great potential to be adapted to a proper bedroom or enlarged. Friends of ours who lived on our estate (grade II listed) made all sorts of changes to their flat (a bigger one than ours, but smaller than this one you're looking at) to make it workable for them and their two kids and have no intention of ever leaving.
Downsides were high service charge (ours was about £120 a month, but the flip side is that our utility bills were tiny) and single-glazed windows (a bit rattling and cold in the winter, but bearable). We couldn't do anything about the windows as they were part of the building's listed status.
I miss every day the sense of community we had on our estate. DC1 had friends the same age she would have gone to school with. We bumped into people every day we knew around the estate or in the local shops. There were regular community events and community organisations to be part of.
Plus, it's a great investment. Worst case you live there until the DC are about 8 and then move on. But I think they would have a lovely time growing up there.
If you don't decide (or miss out on) the flat, I'd keep looking. The house looks a bit grim to me!