Alright, quick and oversimplified crash course in network architecture: If you want to build a large scale network (and I am not talking about 2 xboxes but a space station, the size of Manhattan x 5), as long as bandwidth isn't an issue, the most economical way to build, operate and maintain a network is a star shape. This means you build one big mainframe and connect every system and terminal to that central location. This has heaps of advantages, I am not all going to go into but I guess it's fairly obvious that this makes it easy to operate the network and organize everything. There are other forms of architecture which make things a little more complicated but ultimately, for the purpose of what we are arguing about, they don't offer a much different scenario and this one is the easiest to explain.
Now, imagine you want to isolate a system from the rest. what you need to do is to build two mainframes, one for the isolated system and one for the rest. It is even likely they did this for the Citadel because they wanted to keep some stuff secret (the catalyst, the relay functionality, the relay network remote control, the receiver for the keeper signal, etc.). So they definitely need two mainframes, one public one and one secret one.
You need to put in some extra planning to build this properly and not have accidental connections (you need parallel wiring and such), you even need two power sources (if skynet really wanted to play Halo badly, it could have probably just hacked your xbox through the power line and you and your friend would have had to sit out the apocalypse playing checkers). So that's extra effort. If you want to now divide the secret catalyst from the secret relay functionality system, you need to not double but triple everything. That's where the "extra effort" comes from and if real life large scale networks are even remotely comparable, it's a lot of extra effort. .
But as I said, it's probably not a biggy for the catalyst and the reapers to do this 'cause they are super machines. The real question is (again): Why would they? Even if we assume your head canon that the catalyst wanted all of these events to play out as they did and never interfere (an idea which would hilariously backfire if I choose destroy as I have explained), it still wouldn't hurt to build a normal star network where everything is connected. It's not like the catalyst would have to use it. But suggesting that they planned this whole thing out more complicated than it needs to be in order to deliberately keep the catalyst from these systems because he knew that he'd never need them (which is wrong) is just preposterous.
But then again, since the Shepalyst can suddenly open the arms in the control ending, it seems that all of this stuff (even public and secret functions) are connected somehow after all anyway so it makes even less sense but I have no good answer for that one.
Now for Vigil: Vigil speculates a lot, that's true but AFAIK, he doesn't outright lie (and has no reason to either). He also always tells us explicitly when he speculates and he doesn't do so in the following: Vigil states :"[the reapers] gained control of the Citadel and through it the mass relays."
Now you wonder how Vigil can know this and it's a good question. My best hypothesis is that he still gets some data from the Citadel through the conduit (we know that e.g. comm boys can communicate even through relays when they are not active (spinning), so I guess Vigil can do the same with the conduit. This would give him all the info he needs.
Vigil continues "Communication and transportation throughout our empire was crippled. Each system was isolated from the others, making it easy pray for the reaper fleets." This indicated that the Citadel controls the entire network, not just the relays around the Citadel. However, I missed one thing here: I always assumed that from the Citadel, you can control every relay individually, turning some off and leaving others on. This would have given the council an enormous advantage in ME3. They could have isolated the reapers while still allowing the allied fleets free movement through the part of the network where there are no reapers yet. However, this is not explicitly stated, so it may be that you can only turn the network on or off as a whole. And you are right, that would be a good explanation for why the council doesn't use this function.
However, even if that is the case, it would still have made sense for the reapers to attack the Citadel immediately in the beginning of ME3. As you said, even closing the entire network would give them a massive advantage.
And to lock down the relays would only hurt organics more then it would effect the Reapers. They are capable of travel across the vast distances of space without Relays. They are not able to.
As for your last paragraph:
1. and 2. I agree it's far fetched but quantum shielding may be different in this regard and loosing the citadel (keep in mind that the catalyst is in there) would be a big deal. And again, this is just a band-aid to explain why the reapers might not attack right away. I am aware it's not exactly the epitome of masterful world building.
3. The Citadel may be different in that regard as well, indoctrination hardly works always the same way but hey, ok, you were the one who asked why it might not work in the first place and this is an argument why it might not, so you answered your own question. And the people in arrival didn't act strange? When I first played it, I was completely weirded out by Kenson and her response to inquiries about indoctrination. She just deflected me. The only issue there was that I didn't get a dialogue option to follow up. Once arriving on project base things got only stranger with her outright revering attitude towards the countdown timer. Also, the Cerberus guys in the derelict reaper (since you brought it up in the previous sentence) are even weirded out themselves as it starts. Good luck keeping all that contained when you do it to a couple of million people who are in constant communication to the outside world. But yet again, I just want to point out that here again you seem to shift your arguments, just to be able to have an opposing viewpoint. I don't mind being challenged but I really wonder what your actual opinion is sometimes.