In my e-shops there may be situations when a customer sent me coins but forgot to add secret order pin code in the memo. In such cases he'd contact me via email and claim that a certain transaction was the one he sent, and then I'd verify it for myself and mark his order as paid.
However, if there were very few transactions, 1-2 per day, it might work. But if there were dozens, a customer might try to cheat me - "that transaction was sent by me, not by that customer who claims otherwise, he didn't send it, I did".
How would I ask a customer to verify his ownership of a certain address then, in order to verify that a transaction was, in fact, made by him? And without too much hussle such as calling, screen-sharing and similar things.
I don't consider using third-party paid services. A public blockchain explorer wouldn't fall in this category.