I'm not missing it, you are missing it. Yes EU companies get a tax refund and US companies don't, because there is nothing to refund when nothing was paid.
>But, the US primarily taxes businesses via corporate income tax (~21% federally, plus state taxes)
Believe it or not, but corporate income taxes in Europe also exist. I can see the argument that tax burden for companies in US is higher because government collects more in income tax and less in VAT tax. I'm not sure that is actually true and strongly suspect it's not, as EU is anything but a low-tax jurisdiction. If anything, that's a good argument for US to introduce VAT tax.