> Both name and surname are connected to your centralized exchange.

But that information is not directly visible by merchant. Needs chananalysis, which gives some uncertainty, unless you paid directly from the CEX.

But CEX is not a requirement to use Bitcoin, I have seen totally computer illiterate people, not geeks, buying Bitcoin with cash peer to peer.

Both name and surname are connected to your centralized exchange. 

If you want to insist that potential layer of indirection makes any tangible difference, then you are arguing semantics, not realities. 
Besides, your merchant most likely captured your name and surname during the payment process too.

In either case, stating that paying with Bitcoin let's you stay private is just blatantly false.

unboiled

How so? How can credit cards be more private than poorly used Bitcoin if there is literally your name and surname sent to the merchant?