Strictly speaking, the actual effective VAT burden depends not as much on your level of income but on your level of VATable consumption compared to your savings. If you are "affluent", but your savings are negative, i.e. you finance your consumption by credit, than the effective VAT rate that you pay may be much higher than that of a poor person that hardly makes ends meet but borrows nothing. This is because you pay VAT both on your income and borrowings that you spend consuming stuff.
Historically, however, rich people tended to save or invest (in VAT exempt financial and other assets) a lot, while poor could hardly afford any savings. This made the regressiveness argument hold true, in general.
Note that the absolute amount of VAT that a rich person would normally contribute would be much higher than that of the poor person, regardless of the level of savings. One of the advantages of the indirect taxes like VAT is that they are more difficult to circumvent compared to income taxes. This is because they are remitted to the tax office (IRS) by the agent (producer), rather than the tax payer (rich or poor person), and because of the producer's perception that the tax is at least partially borne by the consumer. Actual levels of tax lost to avoidance/evasion tend to be much lower for VAT than for the income taxes.
Yet, VAT has serious disadvantages. It may kill certain businesses, especially small ones. Russia at some point introduced a "small business" company tax regime, whereby companies earning below certain threshold (currently circa $2m p.a.) are exempt from VAT and other taxes and pay a 6% turnover or 15% profits tax instead. This regime is now very popular, and companies, as they grow, do fear that they may exceed the $2m threshold because for them this could mean death from the VAT strike.
By the way, one important reason the VAT was proposed by the French academics was that they tried to get rid of turnover taxes as the latter were viewed as leading to monopolization due to their self-multiplication effect at all stages of the production chain. But in practice it often works not as intended - there is a bunch of various exempt entities like banks and other FS institutions, that distort that anti-turnover argument.
Another thing about VAT is that it is complex, time-consuming and difficult to administer, real headache actually. Sales tax is much easier to administer and its administration is much cheaper on the economy-wide scale.
In general, the producers are not always able to pass the VAT cost on to the consumer. The fact that they were able to do this in Britain evidences that there could be some hidden inflation pressure and that the VAT hike affected all producers simultaneously. The ensuing price hike, seemingly, concerned primarily products whose price elasticity is low, like basic foodstuff. For these products it is especially easy to pass the VAT cost to the consumer.
In order to counter that, and to counter the general regressiveness of VAT, basic vital supplies (and services), like select foodstuff items and medical supplies (medical services), are often taxed at reduced rates or altogether exempt. This kind of exempts the poor from VAT, and makes more difficult to pass the VAT costs on to the consumers, as the consumers have a choice of altering their consumption patterns towards VAT-reduced-rate/exempt items.