As pretty much everyone knows now, the Second Vatican Council did not abolish Gregorian chant. On the contrary, the Council required that Gregorian chant maintain pride of place among all forms of liturgical music. This has been at best ignored for decades and, at worst, lied about.
Many groups are laudably trying to revive Gregorian chant. Some are using the support of organ accompaniment.
In regard to chant with organ, and related compositions in that style, there is a new website which is garnering a certain measure of support. I refer you to the Chabanel Psalm Project.
They claim some 700 liturgical scores for Responsorial Psalms, Mass parts, and Antiphons on their site. Generally speaking, they are all based on Gregorian chant.
I am personally not a fan of chant with organ. Chant is just not that hard once you get into it for a while. I have heard chant with organ in many places. The principle reason to use organ is to keep people who can stay on pitch… on pitch. That is one of the most painful things about poorly sung, or sung under the direction of someone who doesn’t know how to work to impress the need for pitch and intonation, actually listening to the chant being sung as you sing it.
Some argue that chant with organ actually becomes a new form of music. Well.. maybe so, I don’t know. I still prefer Gregorian chant without accompaniment.
On the site you read (and these are edited excerpts with my emphases and comments:
Verily, one might ask, “Why accompany the Gregorian melodies at all?”
Three possible answers follow:
(1) This is, simply, a very common practice. [So are guitars and pianos for "On Eagle’s Wings" and "Gather Us In". That doesn’t make it good.] If you ask people why they eat meat inside bread it never occurs to them to give you a history of the sandwich. They simply do it. [A truly silly argument.] … Several authors claimed to hate this practice, but claim that they were forced by the sheer commonness of this practice to publish their own methods for accompanying chant! … [So, the argument ehre is "Just give in because lots of people do it."]
(2) Under certain circumstances, organ accompaniment aids the singer. [This is clearly true and it is a good argument as far as it goes. However, it doesn’t say much about the desire to improve the singers and work toward unaccompanied chant.]
(3) For some chant (not every single piece in the repertoire) a well executed organ accompaniment makes truly gorgeous music. It sounds quite different than accompanied chant, but it is beautiful in its own way. [Okay… this is a far better argument. But effectively, you are arguing a change or modification of genres.] Comparing well-done accompanied chant to unaccompanied chant is like comparing a beautiful lily to a beautiful rose: pointless! [No… I don’t think it is quite like that. It is more like comparing a "historic" variety of rose with the "modern" hybrid varieties of roses. They are different, but clearly closely related.] Both can be so incredibly beautiful. Oh, let us praise our Savior for both! [Well… perhaps the choir director.] The happiest memories of my life consist of listening to chant (mainly unaccompanied) for hours and hours, day after day, month after month, year after year. I can think of nothing more beautiful in this entire world than Gregorian chant. It is so beautiful words cannot describe it. However, this does not mean that chant with a good accompaniment is not beautiful. They are both beautiful. All hyperbole aside, [Thank you.] both are so beautiful that I am fainting [!] just thinking of the beauty. May God be praised for his creations! Domine Dominator noster quam grande est nomen tuum in universa terra!
You get the idea. He is in favor of Gregorian chant, and with organ accompaniment.
In any event, you might look at that site, which is quite interesting. There are links to historic editions, which are fascinating. You will want to look at his material on the different schools of accompaniment.
There are some audio clips on the site, but I couldn’t for the life of me get them to produce any sound I could hear.
The bottom line is that this site is promoting Gregorian chant. That is a laudable project. Take a look.
It could be very useful for parish musicians.