Yes, you are completely wrong. This is rather how sales tax works.
With VAT, you do pay it on every link of the chain, but then businesses are allowed to recoup it on their purchases, while end-users are not. For example, if you are making and selling chairs, you pay VAT on the wood you are buying, and your buyer is also paying VAT on the chairs they buy from you. However, you are allowed to deduct the VAT you paid for wood from the VAT owed to the government by selling chairs.
To make it concrete, suppose you buy wood for $1000, and sell chairs for $2000. Suppose VAT rate is 22%. Then you actually pay $1022 to wood wholesalers, and sell your chairs for $3066. Your sale of the chairs makes you owe $66 to government, but the invoice you got from wood wholesalers allows you to deduct from your tax bill the $22 you already paid when you bought wood, and in the end you only owe $44, which is exactly the 22% of the value you added ($3000-$1000 = $2000).