I mean this with no disrespect, but can someone please explain what is so spiritually beneficial about watching a procession? I must say, I don't get all this talk about the urgency to watch the procession. I've never found them terribly interesting.

I'm somewhere in between Gavin and the proponents of watching. Which is to say that I have never had any difficulty at all in doing both - singing And watching; and have found that one contributes to the appreciation of the other. There is much to be said in favour of the choir singing the introit as they ought; but, if a perfectly licit hymn is being sung, those who choose to stand there and gape cannot fall back on the notion that their silence is ipso facto some variety of profound spiritual exercise. For some, it may be so. But for most, it is likely nothing more than 'I don't want to sing, I will just stand here and watch'. Perhaps the procession should enter in total silence and the choir, too, should just watch. (Why should They be deprived of this great benefit?) I would be among the first to afirm that a hymn should never preempt a part of the mass proper or ordinary. But - hymns Are licit, and not to participate in them is to deprive oneself of the great benefit that comes from actually singing a wisely chosen text, and it is deliberately to refuse to take one's own part in the congregational act of praise and worship through song. For most people not singing is just That: not singing!

At the Potifical Mass, the Introit was sung after the Procession, not at the Procession. People who don't want to sing and don't want to be distracted in witnessing the Procession should be allowed to do so.
Many people in pews in our local parish already told me that there are too many songs they have to sing in Mass. And when my friends came to Traditional Mass, they were so relieved that they don't have to sing so many songs and get distracted by trying to sing all those songs, (because they've been told by the pastor and music directors that if they don't sing, they are not participating), and finally be able to focus on Mass. Many people in pews have to miss other important parts of the Mass, because they have to bury their faces in hymnals and struggle with songs. If hymns are optional in Mass and not required parts, according to the Chrurch's instruction, people should also have an option to sing or not. And no Catholics should be forced to sing the parts that are not part of the Catholic Mass.
If one wanted to have more congregational singing, why not help them to sing more their parts of the Mass, "Responses" and "Ordinary parts" (especailly those in Latin as the Church instructed). They will be able to memoriaze them in time and won't need hymnals soon. And once they learn, they will be glad that they can sing the parts of the Mass, and truly participate in Mass, not just participate in singing. Whether hymns are licit or wisely chosen, hymns have been overused in Mass, and the Catholics in pews start to feel that more and more. I think hymns and other songs can be sung more in adoration or in other times outside the Mass.

Processions?
Perhaps words will always be found wanting as we try to express the marvels of God through liturgical forms. Words themselves may be cleverly arranged and seem to posses the power to shape an infinite number of homes for ideas. Yet the mechanism and the mental processing of language creates some drag. Our daily use of language is very flat and scientific: sort of one-to one relationship of word to idea. (see the quote from Balthasar below). The kind of uninspiring, deflated vernacular language of a recipe book has wrapped our liturgy in a prison instead of a castle! And this Kind of energy does not stimulate the moral center where beauty, truth, and goodness interpenetrate. Have you ever attended a Mass where a very chatty well-meaning celebrant inserted explanations for every gesture? I have found that a distraction. Have you attended liturgies burdened with aphorism and allegorical signs, amateur symbols like packets of flower seeds, sand, and burlap, inclusive quotas, and above all the “theme for today’s Mass? Too many uncrafted words.
John Paul II said in 1998
“If subconscious experience is ignored in worship, an affective and devotional vacuum is created and the liturgy can become not only too verbal but also too cerebral.”
Recently I attended the Corpus Christi celebration where the procession! moved me to fear and humility. At first I assumed the decorations and the expanded liturgy was meant to impress or celebrate the presence of some visiting cardinal. There was a 21 gun salute, rose petals raining from the ceiling, tympanis, and heraldic trumpeters in the church towers. I was scanning the sanctuary looking for some prominent cleric. - But everything fell into place as my eyes settled on the monstrance, the canopy and the massive procession drawing close to me. See more images of the Corpus Christi procession
here.
The procession was immense and gave appropriate consent to the real resence of our Lord.
“The reduction of a knowledge of the truth to a purely theoretical kind of evidence from which all living, personal, and ethical decisions have been carefully excluded entails such a palpable narrowing of the field of truth that it is already robbed of its universality and thus of its own proper essence. If truth and goodness are both really transcendental properties of being, then both must interpenetrate each other and every exclusive juxtaposition of their respective realms can only lead to a distortion of their mutual essence” .-Balthasar

To answer Gavin: I am thinking of the procession at a solemn Mass, in which there are cross-bearer, thurifer, acolytes, lectors, (sub-deacon in EF), deacon, master of ceremonies, assistant priest, celebrant, and if he is bishop, extra deacons, masters of ceremonies, mitre bearers, etc., each vested in garments which are beautiful in themselves, but also which represent the hierarchical nature of the liturgy to be celebrated, and thus the church which celebrates it.

But what is moving about a procession is its motion—ideally it moves from the sacristy down the side-aisle and then up the center aisle, symbolically moving through the congregation, as if coming from it, and moving to the focal point of the Mass which is the altar, marking the altar as a holy place by incensing it, going to the chair for the completion of the entrance rite by singing in dialogue with the congregation the penitential rite, and together with the congregation, the Kyrie and the Gloria, and concluding the entrance rite with a collect. I think that it is the purposeful order in motion which is moving about such a procession.

I came to this conclusion, not in the church where my choir sings, where I have to direct the introit and mainly never see the procession, but when visiting England and attending solemn liturgies there (especially in Church of England cathedrals, where the ceremonies are done beautifully). The congregation is asked to sing a hymn, sometimes even a text from one hymnal sung to the tune from another hymnal. I would dutifully get out both hymnals, keeping the program of the liturgy up to be sure to get the page right, balancing all of this together, and looking back and forth between hymnals to match the text to the tune, confidently singing the hymn, only to realize that I had completely missed seeing the procession. Next time, I left the hymnal in the rack and watched the procession, which was much better. I concede that many of the congregation knew the hymns well enough that they were not as distracted as I (in fact, the reason fro singing from two different hymnals is sometimes that they use the same tune for several different texts, and that tune becomes quite familiar).

I do not I propose that it is an issue of any great urgency. Rather, it is a matter of a proper balance in the liturgy—Mass propers generally accompany another action; the congregation's proper participation is either to move in a procession or to witness others doing the same. It need not be their function to provide the music which accompanies the liturgical action. Shouldn't they see and appreciate the sacredness of the altar which is enacted by the incensation, rather than singing the music for it? The congregation's proper role in singing comes when the singing is the liturgical action itself, such as at the Kyrie and Gloria, as well as the penitential rite. This balance between choir and congregation, proper and ordinary in a Mass sung in Gregorian chant suggests that more effort should be placed in the cultivation of the congregation's singing of the ordinary as their best function.

This is not to say that hymns should not ever be used; a parish with several Masses in a day will certainly need to use hymns at some of the Mass. But I think that at the highest form of the Mass, the proper chants (which the hymns so often replace) should take a priority.

Yes, we English do very much like our...'one song to the tune of another'. It's one of the benefits of metric hymnody, which our predecessors understood much better that I think we do today. I find it particularly useful for singing the office hymns, many of which are Common or Long metre.

Processions are wonderful: we witness Christ leading His people to the altar, where He will be sacrificed, that we may have the chance to know the Father and be with Him in Heaven. This happens three times symbolically throughout the Mass: Entrance, Offertory, and Communion. Coincidentally, these are three great times for the PIPs to put the hymnal down and pay attention to Christ.