At long last, I have completed a WC! I did it as Austria, using transfer occupation/demand unlawful territory/grant electorship during war to vassalize the HRE in the 1480's. I then made sure to kill the colonizers, and then very very slowly protectorated America/Africa/Asia and conquered/vassalized the rest. The transfer occupation feature has made vassal feeding so much easier, since I could dump the overextension on my HRE vassals. The Imperialism CB is also great, since it again reduces the warscore cost (this plus 75% admin efficiency allows you to reduce giant countries to a single province in one war or annex all but one country of a multi-nation coalition) and reduces the DIP cost to 0.
I wasn't careful enough in Africa, so Morocco was able to colonize the province connecting them to West Africa, causing several West African nations to westernize and forcing me to vassalize them. I was fearful the same would happen in Asia, since Kazakh and Oirat took Economic ideas 5th instead of 6th, but I transfered uncoreable provinces to them, which would have prevented them from westernizing, even if they had reformed. Also, I was surprised that the South American OPM next to my West Indian CN did not westernize, but I found out that Natives need to border a EUROPEAN nation, not just a Western nation, to reform their Native government.
I almost missed the Siberian country of Chukchi, but they revealed themselves when they tried to take advantage of my war with Japan to invade Ryukyu. Ming colonized parts of Indonesia and South Africa, but I had made my South African provinces into a trade company, preventing them from westernizing.
Another neat trick: If you're at war with multiple Asian countries, say a small one and a giant one, you can protectorate the giant one by just 100%'ing the small one and occupying the capital of the large one. This was helpful in protectorating Malaya and Ming.
Vassal Shout-outs: Guyenne, Provence, Bar, and Normandy for partitioning France; Navarra, Genoa, Galicia, and Granada for partitioning Iberia; Tuscany for taking over Italy and parts of Egypt; Byzantium for taking over the Balkans and parts of Turkey; Genoa for taking over the Crimean, Caucasian, and parts of the Persian regions; Milan and Savoie for controlling Arabia; Mantua for controlling the Red Coast of Egypt (they never cored any of those Egyptian provinces, even though their DIP tech should have put them in colonial/coring range...); Bohemia, Brandenburg, Pommerania, and Livonian Order for partitionin Poland, Novgorod for becoming the new Russia, Bohemia for helping me partition Hungary, Tlemcen for replacing Morocco, and Kazakh and Oirat for forming the anti-westernization wall in Asia.
Without further ado here are the screenshots:
Here's the Old World before forming Greyskin: Gujrat became Milan's protectorate after I gave Milan Persia's last province, so I had to unite the HRE to get the WC achievement.
Here's the New World before forming Greyskin. Friesland was colonizing Texas (they took Exploration as their 7th idea group), so I had to break 2 truces to finish vassalize 2 African nations before Friesland caused some North American nations to westernize.
The player mapmode also colors my subjects, and apparently also colors my subjects' subjects.
Here is the full extent of the HRE before it was united and dissolved. By this point, all my European vassals were either in the HRE, including the Papal State and Byzantium, making the HRE actually Roman.
Here's my income before uniting the HRE. The vassal income is what it's supposed to be, but thanks to a bug, I was making less than 100 ducats from my vassals.
Right before forming the HRE.
I get the WC achievement the day after I press the button. Unfortunately, this event blocked the map at this important time.
Here's the Old World with the advent of Greyskin. It would have been better if I annexed the Ottomans and Egypt, but they hated me too much to be diploannexed in time.
As you can see, the Reformation was almost nonexistent, even though passing one of the later reforms claims that Princes can choose their religion. Guyenne went Reformed (and somehow lost their center of reformation), while Great Britain and Cornwall became Protestant, but were converted when those states were annexed (the COR in London magically disappeared without being converted, preventing me from getting the Center of Attention Achievement).
If you unite the HRE, you get all those provinces with the same autonomy they originally had, so 0% instead of 50% or 75%.
My trade income skyrocketed after annexing all of Europe and sending my 3 merchants to collect from the 3 European end nodes.
I also got every single trading bonus after forming the HRE!
In addition to the WC achievement, I also got My True Friend, Vasa or Wettin, Total Control, and That's a Silk Road. Being a minion master and watching my 1 million+ strong vassal army occupy all the things was very fun indeed, though transferring occupation of tens of provinces at once got tedious, especially since the list of vassals wasn't sorted the most optimally. Making sure nothing in the entire world went wrong was also annoying, but it was all worth it in the end. Hopefully, this will be the most tedious game of EU4 I'll ever play.
In conclusion, I have made the Hapsburg Motto come true. A.E.I.O.U: "All the world is subject to Austria."