I know I keep saying it, but this sort of means-testing is never fair, in any sense of the word. There will always be people who are equally in need (of whatever is being means tested) who will miss out because they sit just outside of the qualifying criteria, so the act of lifting one group to one side of an arbitrary poverty line pushes another group in the other direction.
If a bill is £5 and the cut-off for help with it is £10, someone with £9 is given a fiver in aid, so still has her £9 when the bill is paid. Someone with £11 gets nothing, so is left with £6. It doesn’t matter whether the cut-off amount is based on income or savings - it is manifestly unfair. This sort of thing causes resentment against claimants (which plays into the government narrative of the undeserving poor) and traps the people with £11 in poverty - whatever they do (save, work longer hours, even get cash for their birthday) to try to improve their lot is futile, as they are deemed by others not to be in need by dint of having found that extra money, so they will stay bumping along the poverty line.
It’s not just bills, there are so many things that people one one side of a line get free that those on the other do not. Entrance to events, prescriptions, school meals and so on. It would be so much fairer if we all paid more tax but all got the same subsidies, whether someone with no idea of our individual circumstances deemed us ‘in need’ or not.