It’s not an argument, just a clear pointing out of people calling themselves evangelicals, but with different opinions on other topics.
your own belief system differs from other people calling themselves evangelical. So the “No true evangelical” argument fails as well, every evangelical has their own belief system - down to the Oneness Pentecostals rejecting the Trinity
Rather, a “No true evangelical” argument presupposes a criteria for being so, and which is what you have asserted is some idea of being born again. Thus it is you which are engaging in a “No true evangelical” argument since you exclude those from being so, but which you fail to substantiate as being held by this pastor whom you call evangelical, and to show why this is criteria is valid or unScriptural.
And oh yes, the fallacious unity argument by a member of fractured church in which even proabortion prohomosexual public figures are counted as members in life and in death, while self-proclaimed "true Catholics" excommunicate them and each other based upon their judgment of what true Catholic teaching is and their interpretation of it.