Try running this experiment. Place a mic about 6 feet away from you. Record yourself with the mouthpiece and then with the tube. While you may hear a difference while you are playing (accordion timbre versus harmonica timbre as you noted in your post), I suspect when you listen to the recording, you will not hear much of a difference at all. In others words, the audience hears pretty the same timbre whether you are playing with a mouthpiece or a tube.
Another way of looking at this is, if you close mic some melodicas and make recordings from the top, bottom, left side, and right side, and then listen to the recordings, you will hear a notable difference in timbre between the recordings.
Hence, when you play with a mouthpiece the sound impinges on your ear in a certain way from a certain angle and you hear a certain timbre. As soon as you play with a tube, you are now hearing the melodica from completely different angles and distances. Hence you think the sound of the melodica has changed. And it has as far as your close-up hearing is concerned. But when you get 6 to 10 or 15 feet away from the melodica, these changes in timbre are much less notable as experiment one above will show.
At least those were my findings when I ran into the same observation as you.
I played my Hohner Piano 36 for months with no mouthpiece. Then one day I put on the 3.5-inch straight mouthpiece. I was immediately struck by the difference in timbre. I could not believe that a 3.5-inch mouthpiece could make that much difference in timbre. I could not figure it out so I ran experiment one above. I found no difference in timbre on the recordings. Yet my ears heard a difference when I played. Moving the melodica 3.5 inches away from my ears made a big difference in timbre! Seemed hard to believe but experiment one supported my hypothesis.
The differences you hear when close micing (Is there such a word?) show how important mic position is when close micing. Record a Piano 36 from the left side and right side and you might think you are listening to two different melodicas.
That is my story and I am sticking to it. 🙂 For now.