My last camino, my intention was to avoid that day - crowds and such stress me out. But in the last couple of weeks, we were on pace to make it on the 24th, so we went for it. The plaza that evening is a madhouse, and barricaded in - once you're in, you can't leave, and no bathrooms. Instead, we went up to Alameda Park - which also hosted a carnival, with music, street food and rides - and sat on the hill to watch the fireworks over the cathedral.
I have been working as a volunteer at the Pilgrim Office in Santiago for the past several years around the feast of Santiago on 25 July. I can tell you from experience that you MUST book a room in advance.
If you show up in town without a confirmed reservation, you will NOT likely find a room anywhere near downtown. You MIGHT be able to find a space at Monte de Gozo, or the Albergue Seminario Menor (out of town), but everything inside the old town will be GONE.
Consider that the Feast of Santiago is a big deal because it is both the feast day for the city's namesake; PLUS, Santiago is the patron saint of the entire nation of Spain. So, they pull out all the stops. The fireworks go off at midnight on the 24th, leading into the official feast day on the 25th. Shops and services are, for the most part closed. Cafes and restaurants are open. Markets are NOT.
The city is mobbed, both with pilgrims who want to receive a Compostela on the saint's feast day (Yes, the pilgrim office is open), and hundreds of thousands of tourists, politicians, and the necessary security and police folks. Further, if the King of Spain decides to participate in the High Mass, and related activities, the number of visitors is even greater.