I was just listening to a talk that mentioned why specific names of God are used in the Bible.
God (Elohim) is used when you’re talking about God from the point of view of Gentiles who aren’t under the Covenant, the whole world, etc.
The Name or its substitute, Adonai (YHWH or the Lord) are used when you’re talking about the Covenant, Jews under the Covenant, etc.
Father is more a name for stuff that comes under the New Covenant.
(There’s a lot of mix and match, and of course there’s parallelism where both are used, because parallels what Hebrew poetry is all about. But if you pay close attention, you can usually see the shade of meaning that the author meant to convey, by using one term here and another term there.)
So even though Jesus is talking to a church composed of those already Baptized, it sounds like He is making it clear, Bible-style, that His offer extends to people who haven’t been baptized yet and people who are still pagans, even.
Also, Jews in the Bible who say “My God” or “the God of my people” are emphasizing that they are not talking about other people’s random false gods. They are talking about the real true God Who can do stuff, and does do it.
So the people who “say they are Jews and are not,” and/or the Christians who don’t overcome - they don’t really worship the real God, even though they seem like it. Pagans are eligible to be pillars before they are… unless they repent.
I’m sure there are better answers elsewhere, but those are my thoughts.