You missed a bit… quite a bit actually.
VAT is only part of the puzzle. The majority of tax is collected through income tax, not VAT. Those on lower salaries are likely to pay VAT on most of what they earn. However, our tax system is designed to take more the more you earn - not just in £, but also as a percentage of your income.
A lower earner gets a tax-free allowance of c. £12k and pays 20% tax plus NI. Whilst some may not qualify for additional benefits, many others receive child benefit, childcare support, and may also qualify for additional payments and support with housing, childcare, prescription charges, etc. So whilst they tend to spend most, if not all, of what they earn (on rent, bills, food, consumer goods), they may also draw quite a lot out of the system - a system that other workers cannot access.
Someone earning over c. £50k pays 20% up to that amount and 40% on everything over that, plus NI. At a certain point they will lose child benefit, but may still get support with childcare costs.
Someone earning £100k and above pays 20%, then 40% up to £100k, and they also start to lose their personal allowance - £1 for every £2 earned above this until the personal allowance is wiped out - an effective tax rate of 60% in that band. They have already lost child benefit and now they also lose childcare support, and they still pay NI.
Someone earning c. £125k+ is really winning… they get hit with 20%, 40%, then the effective 60% band at the point the personal allowance is fully removed, and then 45% on anything earned above that. They still pay NI but have no personal tax allowance, no child benefit, and no childcare support.
ISAs: the money paid into ISAs is money that has already been taxed.
Pensions: they are not tax free either. The government effectively delays the tax you pay on your pension as an incentive for people to save into one to reduce their dependence on the state in retirement. You can draw up to 25% tax-free, but everything else will be taxed at whatever your tax rate is at the time.
Savings: High earners with a lot of savings generally pay some level of tax. That tax is typically on money that has already been taxed before it is put into a savings account.
In addition to income tax and VAT, higher earners will likely pay higher stamp duty, and may also face additional property related charges on top of council tax (for example where a high-value property surcharge applies, if introduced).
TLDR: Saying high earners pay less tax because they save more is a best being selective with the truth, if we are being generous... VAT is regressive, but it isn’t the whole picture. Income tax is progressive and far more important when it comes to understanding who actually pays the most tax in the UK.