"the people somehow all have brilliant knowledge of military tactics."
I think this definitely was ignored some way along, and everyone seems to be taking it a little too realistically. In storybooks there is room for more heroic bravery than reality. Lets clear the don'ts first.
1.A head on attack on foot would be military suicide for sure.
2.Facebook is already off the cards because you can't arrange an attack on America (which this most certainly would be seen as) using a system that the Police have unlimited access to. That's just silly.
3.All 220,000 cannot be direct attackers.
4.Infiltrating definitely has to make some of the plan but you waste resources if you make it all of the plan. 100 people cannot be sneaky in that small space let alone 1000.
When taken at face value then you have a much, much better chance at success. You have enough people to build and create solutions to the obstacles in an engineering and logistical capacity. Not everyone has to be a warrior and just one of those guys being an accomplished engineer can lead a team to make pretty much anything with enough time. However you swing it this is necessary anyway for transport so you may as well exploit it for more.
I personally, if writing a story, would need to send in at least 15000 active combatants as A51 is big and I expect at least 1000 superiorly equipped troops on standby. I would be looking at ground penetrating radar solutions - homemade radar is very feasible thanks to google and past wars - to image and detail what the force would be up against tactically and locate key substructures like power, communications, servers and power backup. Servers because they likely have UPS batteries to keep from damaging them in a brownout, and that may allow them to still communicate down network lines. The key is to know more than your foe and to slow down response.
Then, with knowledge of all the heavy blast doors and nuclear style defenses, my story would go something like this:
Have the engineering team design/steal a tunnel boring cart on wheels which could start drilling under the base during this. With a longer pointed nose it theoretically could be pretty invisible to the bases radar and being underground aircraft radars would not see it. We'd need a few (say four) coming from opposite directions to confuse the powerful advanced SIGINT that would be present on the base as it gets close. Even if not confused, the staff watching would be unlikely to believe that 4 were approaching simultaneously as it seems illogical. Being a long distance to cover from the nearest accessible underground sewer, this all can be planned and arranged on safe ground and can take as long as they like, but they would need to start the procession maybe 15 mins before it would be heard inside the base. The tunnel borers would need oxygen supplies and would close the tunnel at the entrance with sandbags to avoid the sound permeating the city via the sewers as soon as is possible. Due to being on wheels, simply angling it on a slope may allow it to be run somewhat unattended as gravity can do the pushing into bedrock. This presents a very time expensive problem if the drills are discovered earlier than planned. One cannot simply blow up the desert to uncover them and thus must find the entrance to each tunnel which will be off-base and out of the bases GPR range.
Create a civilian-style distraction outside the base to draw attention. This is a distraction that would have to force some people out for at least a few hours, either keeping the blast doors open or dividing personnel counts. It must be noisy so something like a politically/celebratory/protest motivated march of say, 10 thousand people complete with fireworks and horns. Think Gay Pride or a small music festival procession just over the edge of Military land but as far from the base as possible. Both have the advantage of being politically charged to deal with recklessly (shooting gay people embracing their rights and/or young people is generally thought of as worse than shooting random civilians). The best thing about this is with 220k people it's somewhat repeatable. If it's cleaned up too fast, start another one somewhere else on the land. No Army is going to butcher 10k people just because another group was hostile. They have jobs to keep and these are citizens on US land.
Have the marchers covertly armed and allow the ensuing hold-up that the military would use to devolve into a gunfight to maintain the image of a visible confrontation as a decoy. Be sure to give the image that it's an act of passion and unrelated to breaking into A51. Think Native American blockades of military convoys if you need a pointer.
It is very unlikely that there are over 10,000 people within A51 so after the capture or death of on site staff by a small portion of that march, there would be another non-A51-staff armed response, definitely of lethal intent and probably via armed helicopter or the like. The issue here that things would ride on is that, being civilians in a march/event they are all considered independent of each other. A military force still needs to differentiate between attacker and campaigner. A show of hands and white flag, or the presenting of said hostages at this point would force that helo to land, prolonging things further. Fireworks/Large speakers can retain noise cover.
When the Drills reach the bunker many meters below ground and start actually boring through the bunker, progress will become VERY noisy and slow, as reinforced concrete is tough to drill. Alarms would be raised and it would be of paramount importance that all drills bore the bunker simultaneously to randomise the noise source, with the one furthest from the power source breaking through the wall when there is only half an inch or so left for the second and third etc to drill. Allow capture of first group and subsequent disabling of alarms that would be raised over the noise and vibration, then have group two break that final inch by hand (quietly) and swarm the base (quietly) with superior numbers, closing the blast doors overground and disabling power, comms, and recordings to buy time for them to scalp over the base and allow them to escape. Due to the first tunnel, this should be easier as most within the base are going to be figuring out what to do with that first hole in the wall. Leave via tunnel and use explosives to cover exit tunnels, breaking the other 1 inch thick entrances to leave multiple directions of exit to any would-be pursuants.
Mostly inspired by movies and videogames. Increase complexity to cover anything I missed. Also realise that 220,000 people is more than most countries entire army, including support personnel, so saying it's infeasible is simply not true. With that first line, you basically have a poor but seasoned, massive, religiously committed Private Army, not 220,000 civilians.