Like all caricatures, there is a hint of truth to them. There are times when people will pray the Rosary during Mass in times past. This is almost non-existent nowadays, but there is some benefit at times. To associate with holy thoughts during the Sacrifice of Calvary is never a bad thing. To meditate especially on the Sorrowful Mysteries during this event cannot be faulted.
Yet I believe Holy Mother Church calls us to something far greater. This individual in such a scenario would be "praying at Mass." St. Pius X called us to "pray the Mass." He states:
The Holy Mass is a prayer itself, even the highest prayer that exists. It is the Sacrifice, dedicated by our Redeemer at the Cross, and repeated every day on the Altar.  If you wish to hear mass as it should be heard, you must follow with eye, heart and mouth all that happens at the Altar. Further, you must pray with the Priest the holy words said by him in the Name of Christ and which Christ says by him. You have to associate your heart with the holy feelings which are contained in these words and in this manner you ought to follow all that happens at the Altar. When acting in this way you have prayed Holy Mass.We find in this quote the beauty of the Mass. The Mass is not just the function of the priest and the servers at the altar. If if is that way in our parish and in our hearts, we are doing it wrong. Every person should be truly participating in the Mass. How can one expect the grace of the Eucharist to be fruitful if received by a soul that is not fully involved with the Eucharistic sacrifice?
Many people have twisted this perfectly noble concept however. To them, participation means "doing something" almost always in the vocal and emotive. People are called to clap their hands, move around, play guitars and drums, and do just about everything in the mass as lay people except consecrate the hosts themselves. (Alas; The Holy Spirit places a check on our narcissism here thankfully.) Yet this was not participation to St. Pius X. For St. Pius, active participation implied an interior involvement manifested by the exterior.
Our souls can perceive the symbolism because our eyes follow the symbols. Our souls perceive the wisdom of the words of the prayers because our intellects are focused on the words being said and desiring to learn from them. The soul is lifted up to heaven through the smelling of the incense that rises to the heavens. our souls taste the spiritual sweetness that is the Eucharist through the reception in our mouths. And when these senses are not sufficient, faith elevates above them.
This is the beauty of the Catholic Mass. The Church engages the entire body and soul of the Christian. If our faith should teach us anything, it is that prayer occurs numerous ways. Sometimes prayer involves saying words. Sometimes pray involves doing actions. And other times, prayer involves being completely passive and letting God do His work. From this it follows is that it is the obligation of every Catholic to pray the prayer that is the Catholic Mass. We may not be able to offer the Eucharist to the heavenly altar the way the priest can, but we can offer ourselves towards the Father, and the Cross purifies that offering of ours and makes it truly acceptable to God.
It is through this understanding that the motions of the Extraordinary Form make sense. Whether we kneel, stand, sing, speak, stay silent, these are all part of the prayer the entire people of God offer to the Trinity that is the Catholic Mass.
When we say "repeated" we do not mean a second sacrifice, or that Christ is still suffering on the Cross. Rather, it is repeated in our perspective, bound by time. Calvary is re-presented to us on the altar.