As often alluded to, are the UK’s rich paying less tax nowadays?

The analogy works because top earners are those with mobile skills who if pushed can and will go and work elsewhere.

To a software developer for example it is fairly irrelevant whether his desk is in the UK, Spain or Sydney as long as they have an internet connection.

And btw, you would expect a widening wealth gap to increase the % of UK income tax paid by the 1%.
Put really simply: The number of people in the 1% would stay the same, but as income becomes increasingly polarised the amount earned by each would increase compared to other 99%. So, all else being equal, they would have more income to tax.
That's an oversimplification, but the general principle applies.

If the amount of tax paid by the 1% rises to 27%, or 30%, or 35% (without changes to the tax rules) this would be a sign that things were becoming more unequal, not less.

Plus, a lot of people are left out of your figures. It is quite hard to disentangle what the 1% highest earners actually earn: I can't work out from that link who those 1% are. Are they just the people who pay income tax via PAYE? Because that will exclude a lot of seriously wealthy people.

That 1% own about 21% of the wealth, though which does put the tax they pay into perspective. As for your comments about VAT etc - you are aware that poor people pay that too, right?
If you do count ALL taxes, then those on low wages can often spend a high proportion of their income on tax compared to higher earners.

Even those who pay no income tax pay other forms of taxes, and contribute to the economy in other ways:
Unpaid carers save the state a lot of money.
Many middle class and wealthy tax payers are able to work the hours they do thanks to cleaners, nursery workers and care workers.
Teachers and teaching assistants help ensure the next generation of tax payers has the skills needed to earn money for the country...
I could give more examples.

cd: Your analogy only works if those 10 men are all dependant on each other in other ways. eg, if the rich guy sold his products to 7th, 8th and 9th, if the 5th and 6th worked for him and if the first 4 men worked in the bar/brewed the beer, and depended on tips to do this. £100 for a weeks worth of drinking for 10 men is pretty bargainous, so I expect someone is being exploited somewhere.

There is a story that illustrates how our taxes work

It concerns 10 drinkers in a bar who decide to settle their £100 weekly beer bill roughly the same way we pay our taxes. So, the first four men (the poorest) paid nothing; the fifth paid £1; the sixth £3; the seventh £7; the eighth £12; the ninth £18; and the 10th man, the richest, paid £59.

Then the barman decided to give them a £20 discount for being good customers. The group wanted to continue to pay the new £80 bill the same way as before. While the first four men still drank for free, the other six divided up the £20 windfall by following the progressive principle of the tax system. So the fifth man, like the first four, now paid nothing, making a 100 per cent saving; the sixth man paid £2 instead of £3 (a 33 per cent saving); the seventh man paid £5 instead of £7 (a 28 per cent saving); the eighth £9 instead of £12 (a 25 per cent saving); and the ninth £14 instead of £18 (a 22 per cent saving). The 10th man paid £49 instead of £59 (a 16 per cent saving).

The men then began to compare their savings. “I only got £1 out of the £20,” declared the sixth man. He pointed to the 10th man, “but he got £10 – the wealthy get all the breaks!” “Wait a minute,” said the first four men, “we didn’t get anything at all. This new system exploits the poor.” So the other nine men surrounded the 10th and beat him up. The next week he didn’t show for drinks, so the nine sat down and had their beers without him. But when they came to pay, they discovered they didn’t have enough money between them to pay even half the bill.