Processions are done probably for reasons of pure convenience and historical reasons. A Mass has different people: priest, deacon, maybe subdeacon, servers, master(s) of ceremonies, bishop maybe, cardinal, etc. There can be massive numbers of people, and so a procession just makes sense. What else are they to do, come in at their own pace at different times? That would be slobbish. Be there before everyone else ready to go? A waste of time and unofficial, more like a business meeting.
Processions, when well coordinated and with the requisite ministers, are grand, glorious things. They look good, they smell good, and they give a sense of nobility to liturgy. In the same way a king or queen processes with court, liturgical ministers (excluding, I hope, EMs…) process together with the celebrant. A procession is just a simple way of coordinating things in a becoming manner. We have wedding processions, royal processions, graduation processions, sometimes processions of participants in a court of law, and so on. It just lends dignity to the event.