The main issue with council tower blocks is
a) standard of the regular communal area cleaning and maintenance regime
b) housing people in them who don't want to live in them but have no choice about it.
If you can avoid (b) it's no real problem - so long as they are properly maintained.
Big cut backs in the 1970s in (a) resulted in many blocks getting a very poor reputation far too quickly.
The issue with council housing finance is that it tends to result in maintenance being done in 'big bursts'. The gaps between these big bursts result in decline which in some cases can get so severe that the flats get a very bad reputation which then results in anyone who can move out doing so and only those with no other choice moving in - so it then creates a spiral of decline to which the solution is then presented as just best to demolish them.
Two experiences of interest involve taking a group of visitors round a high rise block that they expected to be grim but were visibly taken aback to find how smart and tidy it was in the communal areas (lifts/landings/stairs etc)
Other being taken around an 18 storey block of ex council flats that had been sold by the council to a property developer who had renovated them and being shown round by the estate agent charged to sell them who extolled how big the flats were (since the more recent comparable private sector new build flats that were the comparator for the estate agent were so much more pokey - the council flats were not exactly big - just built to the standard 1960s 'Parker Morris' size regulations that were abandoned in the 1980s.
There is also an issue (as with any blocks of flats) of social isolation since you don't have a street level view to watch people coming and going to and from (eg neighbours) as you do on a conventional road. This is not much of an issue if you have the time / fitness / money to spend time away from your flat doing other things - but for many in council housing those are not easy options due to low incomes for example, so that can be an issue for some people.
Other issue can be that thermal insulation in 1960s concrete blocks can be poor and they rely often on electric heating which can be costly (gas not seen as a good idea in system built flats after the Ronan Point disaster in 1967ish). This will be part of the reasons for over-cladding with insulation that at Grenfell Tower went so tragically wrong - due to the type of insulated cladding used of course.
And as mentioned up thread not so good for kids IMHO as not easy to access play space etc directly from door of flat as would be the case with a house.
These flats at the Barbican - originally built as council flats by the corporation of London IIRC, must be pretty desirable given the asking prices c£1.3m+ - but that is clearly at an extreme end of the spectrum!
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