Since we already know the end of the Story, let's see where the Church is headed:
Revelation 19:6-10 (ESV)
6 Then I heard what seemed to be the voice of a great multitude, like the roar of many waters and like the sound of mighty peals of thunder, crying out,
For the Lord our God
the Almighty reigns.
7 Let us rejoice and exult
and give him the glory,
for the marriage of the Lamb has come,
and his Bride has made herself ready;
8 it was granted her to clothe herself
with fine linen, bright and pure”—
for the fine linen is the righteous deeds of the saints.
9 And the angel said to me, “Write this: Blessed are those who are invited to the marriage supper of the Lamb.” And he said to me, “These are the true words of God.” 10 Then I fell down at his feet to worship him, but he said to me, “You must not do that! I am a fellow servant with you and your brothers who hold to the testimony of Jesus. Worship God.” For the testimony of Jesus is the spirit of prophecy.
To be clear, in the imagery of the New Testament, the Bride is the united Church and the Lamb is Christ Jesus Himself. So the goal of ecumenism is (or ought to be) the part I bolded: to make ourselves ready for our marriage with Christ.
We can understand marriage imagery: especially if we are ourselves married. The bride and groom prepare intensely for the big day: caterers, officiants, bridesmaids and groomsmen, DJs, guest lists, invitations, venues, honeymoons, etc., and etc. It can be overwhelming. But the primary duty of the bride is the please her future husband and vice versa. In the long run, the marriage will only work if each person commits to giving the other agape love that Paul describes in 1st Corinthians 13.
The Lamb has already prepared Himself: His central task was His work on the Cross. That is done, but He is still active in preparing the Church for her great day. He is ready; we are not.
What does the Lamb expect from His Bride? Well, that seems clear from this passage too: to clothe ourselves in the "fine linen, bright and pure": "the righteous deeds of the saints." Already, we can start to see denominational pitfalls. Since the official canonization process was not in place, "saints" means simply "holy people" or those who have been sanctified. Also, the "righteous deeds" here have nothing to do with salvation: the Bride has already clothed herself with them at that point in future history. We can argue about exactly how that will happen, but we can't argue that it will happen.
John fell to worship the angel, but the angel redirected him worship God instead. His reasoning is that he is a servant of God, just like John and his "brothers who hold to the testimony of Jesus." That's a pretty good definition of the Church historically. It's also a vital part of getting the Church ready: "hold to the testimony of Jesus." The other part is to do what He commands.
What does that mean when the rubber meets the road? As you say: do we try to sing in harmony or do we sing the same tune? Both and more! The songs of every nation will rise up and, though they now sound discordant, will then be harmonious and in praise of One God together. I got a taste of that at Urbana '93 with thousands of fellow students as we worshiped God together and celebrated communion. At that moment, it would not have surprised me if Christ Himself walked into Assembly Hall. Around the same time, John Piper preached these words:
Missions is not the ultimate goal of the church; worship is. Missions exist because somewhere true worship doesn't. Missions is a means to creating worship in the hearts of those who have never drunk from the wells of salvation. The father is seeking people to worship him in spirit and in truth. This is the ultimate meaning of missions—to bring about Christ-exalting worship in all the peoples of the world.
I think the barriers to Reunification are too high. In our sin we have separated ourselves too definitively to be rejoined before Christ returns to set all things right. You and I can never agree on certain issues until we are before God's throne and see everything clearly. But I can look over the barrier and see whether or not God is worshiped in your tradition. (He is.) So the spirit of ecumenism, in my opinion, encourages us to point out the one, true God to each other.
Different streams of the Christian faith have different ideas about how it will happen, but according to Revelation, all will eventually flow into one mighty river. We know this will happen because we are all aiming at the same goal: union with Jesus, the Lamb of God. Ecumenism only makes sense if we are encouraging each other to follow that course.