Don’t be too quick to wish for an organ. Many churches that I have attended that have an Organ also have an organist that drones on the base chords so that the melody we are supposed to sing is undiscoverable. The worst I have heard is a where an excellent cantor was drowned out by another professional musicain, his wife, on the organ. Actually drowned out the Gospel Acclamation.
This is not to disparage the many excellent organists that are out there. But to some the power of an organ is just irresistible. When someone suggested we get an organ or a piano with organ voices, my reply was: First we need to know who is going to play it and how well. It is too easy for an organist to cover lack of skill with volume and slurred bass chords. The human voice is the most important instrument in church. All other instruments should support it, echo the words just sung, etc.
In my opinion, Chant can become so formalized that it is weak. Chant of Scripture should first and foremost strongly express the meaning of the Scripture. If the point made in a line or verse is strong, it should be strongly sung, if hopeful, the voice should exude hope, if sad or pleading as in De Profundis, Out of the Depths, a little pleading and a bit of blues fits. I sometimes find that the music offered for the responsorial psalm verses does not express the mood of the psalm well. Then I adjust it. The Word of God trumps musical perfection.
Again, my opinion is that the piano or organ backs up the singer. It should be loud enough to help the cantor but should be background only to the congregation.
We have a congregation of 120 regularly and 170 Christmas and Easter. We have 6 cantors. By allowing imperfections at first and encouraging them to sing with force and confidence even when making a mistake we have gone from not singing the psalm and Gospel acclamation to never, never not singing them. And we have 6 cantors for 120 active parishioners. Changes and mistakes sung with confidence sound perfectly fine to the congregation since they don’t have the music. The cantors sing solo. It is easier singing in a group and they can develop a better interpretation of the psalm and Gospel acclamation. We are beginning now to sing these parts together, but we spent three years improving individually first. Cantoring solo is as easy as singing in the shower. We sing the Gospel acclamation at weekday masses so that provides a place to practice and gain confidence. It works.
For great English chant, look at the Gelineau Psalms from the 1950s. The music and the meaning of the words are in complete harmony.
I play piano and I am very good. I have played in church since I was a child, and now I am 55.
One year ago, I started taking organ lessons from an excellent, well-educated, and experienced organist/teacher, who is skilled in both concert and liturgical literature. I am a rank amateur (cute pun!

). But I’m getting better.
The problems you describe sound like practice issues to me.
Do you really think that any organist deliberately chooses to drown out a cantor or choir? Do you really think that organists who play for Mass are seduced by the “power of the organ?” Really?
That’s cold.
The organist probably can’t hear the balance well, and if they can hear that there are problems, they don’t necessarily know how to make the corrections.
A teacher or another experienced musician should spend some time in the nave with them and help them choose registration that will be well-balanced and of an appropriate volume. (They will probably have to pay for this, so perhaps if you dislike the way an organist is playing, you could always offer to help fund his/her lessons. I pay $250/month, if you want to get an idea of how much it costs.)
From the bench, we may very well be hearing the music differently than the people in the pews. And even if something sounded good in the practice hour, during the Mass when people have filled the space, the sound may be very different.
And I will admit that I often have no clue how to fix problems. But as of yet, I don’t play the organ when I accompany the hymns or the cantor/choir. I don’t have the skill yet. And when I read posts like this, I don’t find myself eager to play for Mass singing.
Last night I played prelude and postlude on the organ for my parish Masses (a quiet Vierne for the prelude and a jolly Bach for the postlude). Bless his heart, my dear organ teacher was waiting at the church for me an hour before the Mass started, and even though confessions were happening, he helped me to set the registrations so that the pieces would be correct and pleasing to hear. I was very glad he was there, because he understands “warm” stops and I don’t.