As coleopterist said, it's usually a style-guide issue. There's no hard and fast rule about using simple past, simple present, or present perfect. As for the argument that because Singer's piece was written in the past, you'd use the past ... well, that's absurd: everything already said and written occurred in the past. If it hadn't, we wouldn't be able to read it or to have heard it or hear it again (if recorded).
The most important point is to be consistent about tense. There might be reasons for using one tense versus the other two. For example:
If the writer thinks that Singer's argument is still valid, simple present seems a good choice: The argument was stated long ago, but it still holds. The Statue of Liberty was created long ago, but it still stands.
If the writer thinks that Singer's argument is no longer valid, simple past seems a good choice: "Singer argued X, but Rapper proved him wrong in his 2009 article 'X is False'.
If the writer thinks that Singer's argument may still be valid but not 100% persuasive, especially because of rival arguments, present perfect seems a good choice: "Although Singer has argued X and Rapper has argued Y, Crooner has shown that both X and Y contain logical flaws that make Z a viable alternative".
Another reason for choosing the present tense is to give the argument currency and for choosing past is to place it definitely in the past and imply that it's history, not a currently accepted argument. Again, that's a matter of style, not grammar. Grammar can affect semantics, however.
Your argument "...what is written and now being read still has currency[;] therefore[,] introducing an argument for 'argues'" is persuasive enough for me. I'd go with that because I think it's a logical and powerful argument.
There's no reason to insist on associating these tenses with my reasons for choosing them for the example sentences I've offered. They're style choices and expressions of the writer's personal preference in this case, not hard and fast rules. Consistency is the most important rule.