This is somewhat unfair.  An anonymous payment service only has to be anonymous concerning the payment, not concerning the counter service of course, with which it has nothing to do.  Of course, if I want to buy a physical good, I have to provide for some delivery information to the person I'm buying the good from, that's unavoidable.  But that has nothing to do with the payment.

But consider, for instance, that I want to be an anonymous funder of, say, the Islamitic State.  I only want to do an anonymous transaction of value so that nobody can find out that I was the one who has donated value.  Then there's a serious chance that something like Monero can help, while it would be utmost dangerous to do so with Bitcoin.

I would first buy bitcoin on an exchange.  This would be traceable.  Then I would transfer those bitcoins to an exchange that has monero on the list.  That would be traceable too.  Then I would convert my bitcoins to monero.  The exchange would know that.  I would then withdraw my monero.  That's also traceable.  There's no need to try to hide that, it won't work.

But then I would do several Monero-monero transactions between addresses of my own.  If I had a monero node running for a long time, I could do that directly on my monero node for some transactions, and I could use Tor for other transactions.

At this point, I can assume that the trace of the value to me is essentially whiped out.

Next, the most critical step would be to go to a public place, connect to a public net, and, through Tor, using something like TAILS, give the final order to transfer my donation to a monero account of the Islamic State in my example.

I think that that value transfer is essentially unlinkable to me.  In as far as it is, I would say that Monero succeeded in providing true anonymous transfer of value.

Just to make clear: I only use "Islamic state" as an extreme example of where you can think that the powers that be put all efforts to trace funding.  I'm not a funder of those criminals at all.