I think back then it was not even private because you could literally point out the person who made that transaction considering the number of transactions.
A much more important factor is the number of full nodes. I doubt the number of full nodes in first year was even in double digits which means one entity (like a 3 letter agency) could have easily connected to literary all full nodes and see which one is sending the transaction out for the first time and keep a record of that.
Today that is extremely difficult because the number of full nodes is entering 4 digits and there are loads of privacy improvements that have been made in the P2P protocol.