But why can't two companies exchange goods directly without paying VAT? This would make the famous carousel fraud scam impossible and businesses won't have to deal with complicated refunds.
Sales tax in the United States works as you describe. Sales tax is charged only to end customers, not to businesses that themselves charge sales tax. But this means that a criminal business can charge tax and just pocket it unless someone else reports it. They can also evade income tax the same way. Not to mention other issues like cross jurisdiction taxes (e.g. internet sales often evade sales tax).
The whole point of a Value Added Tax (VAT) is that they charge at each level. This creates a system where each buyer reports the tax paid to the seller so as to be able to deduct it. So the seller has to pay the VAT that they charged. Or the tax authorities know and can revoke their VAT license.
If only the end user is charged tax, then fraud is easier than under a VAT. So easy, I doubt they have a special name for it. The fraudulent business just collects tax from end users and disappears. Or simply fails to record those transactions. You could call it missing transaction record fraud, but why bother? It's just straight up tax fraud. The complexity of the carousel fraud arises from the difficulty of evading a VAT.