<<In the book, it was the other way around. It was Adam who gave THEM a speech. The only difference their presence made was for their own personal growth.>>
In the series, the rationale behind this is still kind of there, even if they gave Crowley & Aziraphale more to do in stopping Armageddon. Adam in S1 remains the one who gets it by the S1 finale. His "I've got all the world I need" is the entire ballgame of the series. They have Adam say it more directly to Gabriel and Beez, instead, though, because that's how the story is able to continue past S1 and go more in depths with its themes. It's Gabriel and Beez who ultimately realize that Adam was right and that sets up a S2 of them rebelling against Heaven that leads into The Finale.
<<that personal growth was actually all about accepting that the fate of the Earth is not theirs to shape.>>
If you mean this in terms of they don't have to deal with the whole world themselves because they are just individual people then I agree with you but I feel like it's more like their growth is accepting that we all contribute to the fate of the Earth everyday by accepting that what is within our control is our own, individual worlds. The whole planet isn't our individual responsibilities-- and therein lies madness because you cannot control what happens-- but we can control ourselves and we can help those around us and live as best as we can to reflect the world we want to live in.
If enough of us avoid our own personal armageddons, we stop Armageddon.
<<Any ending to the story that can feel true to me - official or not, I honestly don't care - should lead them to that place, the same one they were at the end of the book, but where the S1 finale indeed failed to take them.>>
It will. The end of S1 was designed to have enough of an ending that it'd work if they didn't end up making a S2 but also only as the first third of a longer story as well. Even if Adam had given that speech from the book to Crowley and Aziraphale more directly in S1, they wouldn't have been in a place for it, especially Aziraphale. That's the whole rest of the story so they pulled back from it in the S1 finale but it's all heading there in The Finale because everything since has been building towards it, imho.