On practice, of course, things aren't this clear cut. Every place I've looked at (not many) refuses to simply deduct the already payed VAT from companies debits, and instead use some complex rules to determine what is "added value" that often enough subsidize large business on the expense of small ones.
Imagine a product which requires two processes, A and B.
Raw materials cost 1
After process A is done, the value of the intermediate output is now 11
After process B is done, the value of the finished product is now 21
If you are vertically integrated: you do A and B, so you collect VAT on (sale price less cost of inputs) = 21 - 1 = 20
Whereas if non-integrated firm One does process A A will collect VAT on (sale price less cost of cost of inputs) = 11 - 1 = 10
And non-integrated firm Two does process B B will collect VAT on (sale price less cost of inputs) = 21 - 11 = 10
So the VAT is calculated on 20 "value added" in both cases meaning that vertical integration gains you nothing at all.Big companies can and do manipulate transfer pricing to minimize VAT payments but this is "playing with taxes" rather than a problem with the concept of VAT.
Thus corporate taxes tend to have an impact across the entire population, somewhat like VATs, see [1]. I feel like this is why they are popular, it sounds like you are taxing someone else to most people.
[1] http://www.pgpf.org/sites/default/files/0102_tax-rates-full....
Investment always lead to some kind of consumption, that is taxed just like plain consumption. I've never seen this taken into account when calculating VATs regressiveness.
Although it can't really be progressive, so the real question here is how regressive is the tax? Does it win over income taxes, that are also regressive?
I mean, if you know food has VAT, you simply increase the amount of money you give for food to cover that tax. And son on: Adjust poverty levels to show that. You can exclude some things, improve public transportation, and other things to lessen the burden.
Of course, the reality is that it is harder to do in some political cultures - the US, for example.