If you divided up a typical class of 30 children and gave them monopoly money in proportion to the actual income of the UK, they would divide like this (this is based on actual income distribution data):
7 pupils would have less than £10 per week
10 pupils would have £10-£15 per week
13 pupils would have £15-40 per week
1 pupil would have £50 per week
1 pupil would have £110 per week
The class needs to contribute together towards £100 to purchase something that they will all benefit from - (I can't think of an example that represents healthcare, education, policing and security and everything else, lets keep it as a simple "something good").
Each pupil also has to contribute £8 per head per week towards lunch (representing the costs of paying for a basic standard of living), and can keep the rest for other expenses - lets say there are some fun things that spare money can be spent on after paying for these basics, representing quality of life over and above basic survival.
Under the current "progressive" tax system:
Of the 7 who have less than £10, the very poorest contribute nothing, those closer to £10 contribute less than £1, so they can still afford lunch but they are unlikely to afford anything fun - they may get something extra once every 3 or 4 weeks if they are careful to save.
The 10 with £10-£15 contribute between £1 and £1.70 each so they have a little bit left over after paying for lunch.
The 13 with £15-40 contribute £1.70-£5.00 and have a bit more spending money.
The two richest contribute £9 and £35 respectively, and have plenty of spending money left over
Under a flat tax system divised to raise the same £100 with everyone contributing the same proportion of income:
The 7 poorest would have to contribute £1-£1.50 each despite the fact that this would mean they could not afford lunch at all and would go hungry
The 10 in the next bracket would contribute a bit more - taking them close to the cost of lunch so they would have much less for other things - the tax would be taking everything they had over and above what they need for basic survival.
The 13 in the next bracket would pay an amount not that different from what they are paying under the progressive system, so they probably wouldn't notice the difference.
The pupil with £50 would pay 80 pence less so would have marginally more spending money (£42 ish left over, rather than £41)
The pupil with £110 would pay £15 less - so would be keeping £90 to spend on the fun stuff as oposed to £75.
If you ran this as a simulation with an actual group of thirty people, any civilised human being who was the richest in the class would feel sickened and disgusted by the idea that 17 people could either go hungry or be deprived of the opportunity to have any quality of live over and above hand-to-mouth survival just so that they could have an extra amount of excess spare money that they don't particularly need.