The practical impossibilities of actually bringing the Korban Pesach

Korban Pesach is one of the most important mitzvas, one of only 2 mitzvos that if not done you are chayav kares. Yet, it is impossible to understand how the Korban Pesach was actually ever brought by a large group of people.

I will start with listing a set of facts based on the Mishnayon/Gemaras in Pesachim and other places:

1. All times listed in the Gemara are based on a 12 hour day 6 to 6 (שעות זמניות)
2. The first Mishna in the 5th perek of Pesachim (48a) states that on Erev Pesach they would start bringing the תמיד של בין הערבים at 1:30PM (7.5 hours) and finish at 2:30PM (8.5)
3. They started bringing the Korban Pesachs only after they finished with the Tamid, meaning they only started at 2:30PM 
4. Korbanos can only be brought until sunset which is 6PM (12 hours)
5. The time alloted to bring all of the Korban Pesachs was 3.5 hours (2:30 PM until 6PM)
6. The Mishna states that the Korban Pesach is brought in 3 groups
7. The Gemara (Pesachim 64) states that they said Hallel while they were bringing the Korban pesachs and they never repeated Hallel more then 2 times
8. The animals were suspended and flayed in the azara
9. King Aggrippas counted the Korban Pesachs one year and came up with 1.2 million
10. The Azara in the Beis Hamikdash was 135x187 Amos
11. The entrance into the Azara was 10 Amos wide

According to the Gemara (9 above) they brought 1.2 million korbanos one year. That means that 1.2 million people needed to go into the Azarah with their sheep to sacrifice it. Lets do the math now.

1. 3.5 hours is 210 minutes, 12600 seconds
2. That is 95 korbanos per second, 5714 per minute. Note: To bring the Korban means to slaughter it, collect the blood, sprinkle the blood on the mizbeach, etc.
3. The size of the Azara is approximately 25,000 square Amos
4. Assuming a large Amah (2 feet) that is 50,000 square feet, Note: part of the azarah was the Kodesh Hakodashim which can not be entered and  there are also the Keilim
5. A loose crowd, one where each person is an arm's length from the body of his or her nearest neighbors, needs 10 square feet per person. A more tightly packed crowd fills 4.5 square feet per person. A truly scary mob of mosh-pit density would get about 2.5 square feet per person.
6. Based on the above (5) 
1. Loose crowd - the maximum number of people is 5000, 3 groups would only get us to 15,000 people, we are 1.185 million people short
2. A more tightly packed crowd - the maximum number of people is 11,111 3 groups would only get us to 33,333 people, we are 1.167 million people short
3. A truly scary mob of mosh-pit density - the maximum number of people is 20,000 3 groups would only get us to 60,000 people, we are 1.14 million people short
7. The entrance to the Azarah is 20 feet wide, meaning at most 8 can come in simultaneously. For 1.2 million to come in would require 150,000 rows of 8. This would require 12 rows to enter a second
8. All the people who come in need to go out the same entrance

Even if we radically reduce the number of korbanos to 100,000 the numbers are still ridiculous

1. That is 8 korbanos per second, 476 per minute. Note: To bring the Korban means to slaughter it, collect the blood, sprinkle the blood on the mizbeach
2. Based on the crowd numbers above (5) 
1. Loose crowd - the maximum number of people is 5000, 3 groups would only get us to 15,000 people, we are 85,000 people short
2. A more tightly packed crowd - the maximum number of people is 11,111 3 groups would only get us to 33,333 people, we are 67,000 people short
3. A truly scary mob of mosh-pit density - the maximum number of people is 20,000 3 groups would only get us to 60,000 people, we are 40,000 people short
3. The entrance to the Azarah is 20 feet wide, meaning at most 8 can come in simultaneously. For 1.2 million to come in would require 150,000 rows of 8. This would require 12 rows to enter a second
4. All the people who come in need to go out the same entrance

Now if we think about Pesach in the Midbar things get even worse. 

1. The population was 600,000 men over the age of 20, giving us a population 2.5. -3 million people
2. If we take teh Gemaras number of 10 people per Korban that would be about 250,000 korbanos
3. The Mishkan was only 30x10 Amos with part of that the Kodesh Hakodashim. 
4. There were only 3 Kohanim to bring 250,000 korbanos

What we see from here is that there is no way that massive numbers of Jews could have ever brought the Korban Pesach. It is simply not feasible. The whole thing makes no sense at all. The Gemara describes what has to happen, yet pays no attention to the sheer impossibility of it all. 

Some will answer, it was all miracles. There are 2 answers to this claim:

1. The Gemara (Pesachim 64) has a dispute between Abaye and Rava if they relied on a miracle to lock the doors of the Azara or not. Abaye says yes and Rava says no. In all disputes between Abaye and Rava (except for 6) the Halacha is like Rava. If Rava says that we can't rely on this small miracle to close the doors, then certainly we can't rely on much greater miracles to allow us to bring the Korban Pesach
2. There is absolutely no source for this. Nowhere does the Gemara say that the bringing of the Korban Pesach was miraculous. And in truth, these would be great miracles indeed.

The bottom line is that based on the numbers it is absolutely impossible for a large group of people to ever bring the Korban Pesach. The short time span plus the small Beis Hamikdash simply does not allow it to happen in any way shape or form.

Update 4/5

There are 2 more issues that I would like to raise:

1. For every Korban that you are Makriv you need to burn the אימורים, the organs on the mizbeach until the next morning. To throw them all on the mizbeach requires 28 per second for the whole night. Of course after throwing on the first n the mizbeach would be covered and there is also the time needed for the אימורים to actually burn. Of course a pile of 1.2 million sets of organs would probably fill up a large part of the Azarah all by itself.
3. In addition to the Korban Pesach everyone who comes to Yerushalayim is required to bring a korban reiyah (Olah) on the first day of the chag. While Korban Pesach applies to a Chabura these korbanot apply to every male, over the age of 13. This means many more Olas Reiyah were brought in comparison to Korban Pesach so if there 1.2 million Korban Pesachs there were at least 4-6 million olas reiyahs that needed to be brought on the first day of Pesach. Again it is absolutely impossible to bring that many Korbanos in a few hours. Additionally, you were supposed to bring Shalmei Simcha as well which adds even more korbanos.

Important note: All the dimensions I gave for the Azarah assumed for simplicity's sake that it was empty and could be completely filled with people, this is of course not true at all. For example, the Kodesh HaKadashim ws 20x20 amah which could not be entered. Additionally the mizbeach was 32x32 amah not including the ramp. So in fact, a significant part of the Azarah was actually not usable space for people greatly minimizing the amount of people who could actually fit into the azara.

Some will try to answer as follows. The Rambam paskens that many people can partake from one Korban Pesach. 

יב [יד] עַד כַּמָּה נִמְנִין עַל הַפֶּסַח, עַד שֶׁיִּהְיֶה בּוֹ כַּזַּיִת לְכָל אֶחָד וְאֶחָד. וְנִמְנִין עָלָיו וּמוֹשְׁכִין אֶת יְדֵיהֶן מִמֶּנּוּ, עַד שֶׁיִּשָּׁחֵט; כֵּיוָן שֶׁנִּשְׁחַט--אֵינוּ יָכוֹל לִמְשֹׁךְ אֶת יָדוֹ, שֶׁהֲרֵי נִשְׁחַט עָלָיו.

A sheep has approximately 30,000 כזיתים so for 2 million people we only would need about 60 sheep. However, there are insurmountable halachic and logistical problems with this idea. 

1. the חבורה needs to eat it together in one house. How would thousands of people do that? In the time of the Beis Hamikdash and the mishkan a building that could hold thousands even hundreds did not exist. See rambam korban pesach chapter 9. 
כָּל הָאוֹכֵל מִן הַפֶּסַח אֵינוֹ אוֹכֵל אֶלָּא בַּחֲבוּרָה אַחַת וְאֵין מוֹצִיאִין מִמֶּנּוּ מִן הַחֲבוּרָה שֶׁיֹּאכַל בָּהּ. וְהַמּוֹצִיא מִמֶּנּוּ כְּזַיִת בָּשָׂר מֵחֲבוּרָה לַחֲבוּרָה בְּלֵיל חֲמִשָּׁה עָשָׂר לוֹקֶה שֶׁנֶּאֱמַר (שמות יב מו) "לֹא תוֹצִיא מִן הַבַּיִת מִן הַבָּשָׂר חוּצָה". וְהוּא שֶׁיַּנִּיחֵנוּ בַּחוּץ. שֶׁהוֹצָאָה כְּתוּבָה בּוֹ כְּשַׁבָּת לְפִיכָךְ צָרִיךְ עֲקִירָה וְהַנָּחָה כְּהוֹצָאַת שַׁבָּת. וְאֵין מוֹצִיא אַחַר מוֹצִיא בְּפֶסַח שֶׁכֵּיוָן שֶׁהוֹצִיאוֹ הָרִאשׁוֹן נִפְסַל. מִן הָאֲגַף וּלְפָנִים כִּלְפָנִים מִן הָאֲגַף וְלַחוּץ כְּלַחוּץ וְהָאֲגַף עַצְמוֹ שֶׁהוּא עֳבִי הַפֶּתַח כְּלַחוּץ. הַחַלּוֹנוֹת וָעֳבִי הַכְּתָלִים כִּלְפָנִים. הַגַּגִּים וְהָעֲלִיּוֹת אֵינָן בִּכְלַל הַבַּיִת:
So tell me where would 30,000 be able to eat the korban pesach in one house? Where would even 3000 people do it? Therefore the practical upper limit for a Chabura is tens of people. 

2. The logistical problem is also insurmountable. Who is butchering the sheep into 30,000 kezaysim? How long does that take? And then you have to distribute the 30,000 kezaysim where and how would that be done in the small amount of time allotted?