I know you're just thinking aloud, but you should really do a bit of checking first. Setting it at £20K would give the top 2/3 of earners a massive tax break at the expense of low earners. It does the exact opposite of what you want it to do.
By your very statement it would help the vast majority of the bottom 2/3rds of earners.
Plus it would disproportionately help the lowest in the group as for example someone earning £20k per annum would see their net salary increase by more than 8% whereas someone on £40k would see less than a 4.5% rise.
I think helping middle earners is beneficial too as they will have seen their salaries hugely affected by Covid. I also think that a boost for middle earners would help stimulate the economy as they would then spend more which would drive greater VAT, corporation tax receipts etc, somewhat offsetting the cost.
Firstly, its not cruel. Most people on benefits gain far more than they pay in tax, so they benefit massively on the arrangement.
Again I disagree it isn't cruel. Taking someone's money when they have precious little in the first place and deciding you know better than them as to what to spend it on, before offering them services (some of which they might not want or need) in return, if not cruel then at a minimum is contemptuous (for example using tax taken from someone who's family have never attended university and are unlikely to in the medium term future to subsidise the wealthiest people in the country to attend Oxford).
The bottom 10-15% at a minimum should be allowed to keep 100% of their own money rather than having the state forcefully sieze it from them in return for services they may not want or need.
This is something I believe in strongly but also accept I'm in the minority. I completely reject the right wing argument that poor people can't be trusted with money (I knows this isn't at all your argument)
Secondly,, its not even illogical. If you accept that we should vary the amount people are entitled to based on their circumstances, then that variation has to be accounted for in either the taxation or the benefits system. To build that into the tax system would create thousands of new tax codes to account for housing need, parental status, health, location etc and would create a completely incomprehensible tax system that would be packed full of loopholes. And it wouldn't obviate the need for a benefit system anyway, because some people have zero income, so you would still need one of them too. So the simplest thing is to keep the tax system as simple as possible, and have the complexity applied only to the benefits system. The fact that it leaves some people paying money out just to get it back again can be seen as a curiosity, but nothing more.
In truth I did qualify that some of my "thinking aloud" was possibly unworkable from an administrative point or view. In a perfect world the tax system would far more straight forward (as would the benefits system) but we do have to work with what we have.
I've always liked the idea of a negative tax rate for example as the poorest people in society often paying an effective tax rate comparable to the highest earners in society (e.g. to earn an extra £1000 net of benefits they need to make £2000 from employment, an effective tax rate of 50%)