To get anywhere, you need to go through a relay. The Quarian fleet isn't just sitting around in Tikkun until called upon for the Banzai charge on Earth. Hackett needs them to split up galaxy-wide and take over logistics, which means they need to empty the civilians out of their cargo holds before they can serve that purpose. They're not meant to serve primarily as a fighting force; they're there to get everything and everyone where they need to go to build the Crucible.
If they split up, your "days to cross a relay" theory no longer holds. And is there any particular reason quarian ships must serve as transports, as opposed to merely relieving other ships that don't have their capacity problem?
The bottom line is this: when faced with a galactic extinction event that will take every resource you have to just maybe not have everyone die horrible deaths, you don't start up pointless wars with each other.
Tali, on why Legion stopped communicating before they invaded: "Perhaps he was trying to fight the Reaper takeover, or perhaps he didn't want to share information with an enemy."
Legion stopped responding to Tali, simply saying "consensus could not be reached." I admit my timeline is partially inferred, but it makes sense to me that the Reaper was already there before the Quarians invaded, and only gave them the code when they were at their most desperate.
We're not really told what Tali was asking for on those communications either, right? And with Gerrel the Destroyer and Xen the Controller on the Admiralty board, Tali being token and Koris being outvoted at every turn, I kind of doubt the negotiations were in much good faith.
I can admit it would be difficult to see them come to an understanding on their own. I think mediation by an outside source they both trust would be needed. Shepard obviously fills that role. But I guess the sequences we got are more exciting to play through than "Shepard sits a table with Legion and the quarians and works out a peace treaty." Of course if Shepard hadn't been a tool and turned himself in, we might've still gotten something better. But that's a whole other can of worms.
"That's definitely Reaper tech. But what's-?"
*Reaper device opens*
"Shepard-Commander, help us!"
Even the timeline you give above requires that the Reaper arrive in-system when the Quarians are already there in strength, without being detected.
I'm sorry, is that too advanced for a Reaper to do?