When the treble starts alone it does not sound like an organ. When the bass starts it likely also would not fool anyone to think that's from the organ...
It is very nicely played, and the church acoustics are helping a lot in creating an organ illusion. Post-processing of the recording could still improve the organ imitation (boosting the lowest bass frequencies for starters)... But what helps most is to use a different accordion (for instance an older Jupiter bayan...).
Speaking as an organist first and very much as an amateur accordionist second, I think a lot of this reveals that the main influence in how an instrument sounds (other than the player) is the acoustic it is played in. You are listening to the acoustic as much as the raw sound of the reeds.
There are many otherwise lousy pipe organs that sound fantastic as they are in great acoustics, and equally many new expensive pipe organs by top builders that leave a lot to be desired as they are in an unsuitable dry acoustic. Organ builders will always tell you the most important stop on any organ is the building itself because it is the resonating chamber a wind instrument needs.
I have a treat for my accordion practice when I move from the soft furnishing living room into a large empty bathroom with solid walls and stone floor: my rusty trusty old pigini suddenly sounds like a top concert instrument!