No, I will not buy if I won the lottery people where I bought online can easily trace me as their winner, it's better that you buy your own ticket online because your privacy is safe, there are people here that offer people for their service and that is buying a ticket for them for a small fee, but what if they record your ticket, they will know exactly that you've won the jackpot.
It could be something to consider and you are just worrying what will happen when you win. But I don't think that it will ask your personal details when buying ticket cause it probably is happening in the time of claiming the rewards. So being traceable as you think is really that far.