The sacrifices were supposed to only be done from the ninth hour until the eleventh (3pm to 5pm, very roughly). That's a pretty tight window; almost 36 every second, almost 2200 every minute!
From this source I picked up a description of the temple passover (different from Moses' and from the current one) that seems consistent:
The paschal lamb was slaughtered in three groups… when the first group entered and the Temple court was filled, the gates of the Temple were closed. A tekiah, teruah, and again a tekiah were then blown on the shofar. The priests stood in rows, and in their hands were basins of silver and basins of gold. … An Israelite slaughtered his offering and the priests caught the blood. The priest passed the basin to his fellow priest, and he to his fellow, each receiving a full basin and giving back an empty one. The priest nearest to the altar tossed the blood against the base of the altar. While this ritual was performed the Levites sang the Hallel [Talmud Pesachim 64a].
I don't know how old the lambs were, but a 50 pound lamb would have something over a quart of blood (6% of body weight), and assuming that only about half of that is taken, and half of that is actually thrown on the altar, that's about 64 cubic meters of blood to drain (and wash!) away. From this site: "Beneath the floor of the Azarah courtyard, near the southwestern corner of the Altar, was a cave. Access to the cave was by means of a 2-foot square hole that was covered by a marble slab. In the floor of the cave was a natural system of drains that led the blood of the sacrificial Altar out into the Kidron Valley below the Temple Mount."
In two hours--call it three because drains can be slow--that's a lot of drainage. I'd heard it said that the result, including the necessary rinsing, would have made blood and water flow visibly, which John noticed in Jesus--a connection I wouldn't have thought of.
If the people stood 300 abreast, with the "bucket brigade" (I'm sorry, that's what it sounds like), that leaves only a few seconds for them to slaughter the lamb and move out of the way for the next people. I think you'd need several rows of people all at work at once. I guess it could work, with good traffic management and thousands of priests.
I remember when a delayed Delta flight disgorged into the Paris airport. It looked like complete chaos at the desks, but somehow we all got processed quite quickly. They got used to this sort of thing, and knew how to manage it.
Still, wow.