> However it is rather uniform and generally it's rate is not affected, e.g. by the total revenue of a business, the type of business, and other such qualities.
Neither are most corporate income taxes. Otherwise the company would use many small corporations instead of one larger one (which they already do to varying extents for other reasons) and avoid the higher rates.
And, of course, the type of business can easily be taken into account with VAT if you like that sort of thing.
> It's very easy to just shift the burden to the customer
It's not any easier or harder than anything other way. Again, if they had thin margins, they have no choice but to raise prices or go out of business, so they raise prices and the customer pays. If they have higher margins they may (or may not) eat some of the tax because the reduction in demand from raising prices might cost them more than paying the tax would. That's the primary consideration in practice, not what you call the tax or whether the customer sees it on their bill.
In some cases the customer may end up paying more than the amount of the tax because the business has to pass on the tax, which reduces demand, which lowers the sales volume they have to amortized fixed costs over, which requires them to raise prices even more.
Though whether the customer sees it on their bill certainly affects the politics of it, because then they notice that they could be the one paying it, even though that was true either way.
> But there's typically also additional taxation on corporate income. Isn't that essentially the "revenue tax" people meant?
It is, the problem with "income" is that it doesn't have a jurisdiction. If you make sales, the customers are somewhere. If you hire employees, the employees are somewhere. If you own property, the property is somewhere. If you make "profit" it's just numbers in a bank's computer, which can be anywhere, so it goes wherever the taxes are lowest without regard to where anything tangible is.
So if you don't want that to happen you have to tax the thing that actually happens in your jurisdiction, and we're back to VAT or payroll tax or whatever.