My feeling about "They Will Know..." is that the explicit message of the first verse proved to be "too ecumenical" (i.e., "not Catholic enough") for the reactionary bunch that came to power in the Church along with Paul VI:

We are one in the Spirit / we are one in the Lord (2x)
And we pray that all (our?) unity / will one day be restored..."

The "we" doing the singing obviously identify themselves as Christians (including Protestants, Orthodox, Copts, etc.), not as a group consisting exclusively of Roman Catholics (members of the One True Church).

I've always felt that the degree of openness expressed in this song led to its fall from favor when the hardliners took charge of the Vatican and proceeded to do everything they could to undo the "damage" done by John XXIII, who they elected only because they didn't expect him to live long enough to accomplish anything...