We have a weekly Scripture Study at my church - we basically never talk about the Psalm. I figure it’s because we all know it isn’t likely to be used - especially since the regular cantor is also a regular attender at Study. We do read it every week and we can talk about it, we just almost never do.
A priest suggested in a talk once that the response in the Responsorial Psalm hands over a key, this huge “hint” concerning the whole set of readings from Holy Scripture. Since talking to him, I read the readings the first time through in this order: Response, Gospel, 1st Reading, Responsorial Psalm, and finally the 2nd reading.
The 1st Reading can be thought of as the prefigurement of the Gospel: “
In times past, God spoke in partial and various ways to our ancestors through the prophets…” (Heb. 1:1). Then the congregation, in the Responsorial Psalm, shouts back, as it were, “Ooo! Oooo! We know where this is going! We know where this is going! This has to do with our lives!! This is where we live! This is where we live!” This is said, not in our own words, but in words supplied by the Holy Spirit over the ages of God’s earlier revelations. To hear what the Responsorial Psalm says–that is, to hear it as a response how the 1st Reading prefigures the Gospel–is to hear it from the point of view of one who “gets it”, who “sees what is coming.”
The 2nd Reading, as often as not selected without any particular intended connection to the other readings, still very often lends an additional but unexpected connection. It is like that ingredient in a particularly fine dish that you just did not see coming. Still, because the “cuisine” is internally consistent, it does work.
Then comes the Gospel of Jesus Christ, the central Word towards which all the rest of God’s revelations have always pointed:
"…in these last days, he spoke to us through a son, whom he made heir of all things and through whom he created the universe, who is the refulgence of his glory, the very imprint of his being, and who sustains all things by his mighty word. When he had accomplished purification from sins, he took his seat at the right hand of the Majesty on high, as far superior to the angels as the name he has inherited is more excellent than theirs." Hebrews 1:2-4
If you look at the readings that way, you can hardly help but see something that the psalm will say in illumination of the other readings. By the very nature of the psalms themselves, it is typically a very personal and heartfelt connection, too. If you talk about what you hear in the Psalms from that perspective, it may be that the cantor will begin to do the same, and you may even hear the Responsorial Psalm sung a lot more often.