It's important to understand why the customer may have wanted SA-516-60 in the first place.
One industry where I see this is for sour service applications where a softer material is desired. I have seen specifications where the customer also adds a note where the maximum tensile strength may not be more than x-ksi above the material specified minimum.
This can sometimes be worked around though. If the customer were to specify SA-516-60 with a maximum tensile say 20ksi above the minimum, you would have an allowable tensile range of 60-80ksi. If you can find SA-516-70 plate with a tensile less than 80ksi, the customer may agree to its usage. This requires some more careful oversight on the purchasing side and checking mill certs.
This is one situation where you have to be careful with such a substitution, it's important to understand WHY they wanted SA-516-60.
(On the other side of the coin, sometimes plant operators are just replacing old equipment and don't know any better...)