The theory that Roderich himself would have been pleased when Anschluss first occurred is linked to the headcanon for which it served as the caption: “Roderich hated his house after World War I. It was the first time it had been empty in a long time, and the silence made him sick.” Roderich had always had other nations around the house: first as part of the Roman Empire, then a link with Vash, then a marriage to Antonio (with other bits of his empire such as Feliciano and Elizaveta around), then just as the Habsburg Empire, then his marriage to Elizaveta, and then eventually he lost everything in the collapse of the Austro-Hungarian Empire at the Treaty of St. Germaine in 1918.
Given that Roderich had really not been alone for such a long time, being trapped alone in a house after WWI would have driven him mad. It is canon that after the Anschluss, Roderich was living in Ludwig’s house. He wasn’t alone anymore. Also, it is stated in Himaruya’s notes that Roderich was in a wheelchair at some point in history. It would make the most sense in our minds if that was during the interwar period, when Austria’s economy was in shambles as, after World War I, Austria only retained “22.3 percent of the inhabitants—6,710,233 persons—and 32.4 percent of the factories of the Monarchy” (Luza 3). Anschluss would have gotten Roderich out of the thrice-cursed wheelchair, which he would have been very grateful for.
So that’s the Hetalia of it all. Here’s a history rant, brought to you by Mod 2. :)
Historically, many Austrians initially welcomed the Anschluss. They had no way of knowing what it would turn into; no one had any real way of knowing. Austria also had its own National Socialist German Workers’ Party (Nazi Party) (Luza 5), that was in favor of unification with Germany. Many Austrians who were not in support of the National Socialist Party also supported “All-German” nationalism (Luza 7). Hitler was charismatic, promised to help Austria’s economy, promised full employment, promised that Austria was going to be something again. He “pretended to bind Austrian wounds and to set the country on its feet again” (Luza 7). He had the support of many young people and idealists who believed that being part of Germany would be good for Austria. Public opinion was, of course, divided, but there was a reason that Hitler’s troops were greeted with flags and flowers when they arrived in Austria.
The vote held on April 10 was almost a month after the Nazis had already arrived. It was not private voting, and no, it was likely not a legitimate vote. There was supposed to be a vote the day after the Nazis arrived, as declared by the Austrian chancellor Schuschnigg, but he resigned before the Nazis arrived due to threats (Luza 7). There was, of course, no real reason for any average citizen to suspect that anything odd was going on. Not when so many were in support of a Greater-German Reich that would raise Austria back up.
We do not in any way mean to ignore the existence of concentration camps and the horrible crimes that were committed by the Nazis there. However, this was long before there was any suspicion of concentration camps; the war hadn’t even started yet. There was no reason to suspect that the Nazis were doing anything untoward here. Nowadays, obviously, it is known that Buchenwald and Dachau are awful places where horrible things happened. But that wasn’t known then, and neither the Austrian people nor Roderich himself could have been expected to understand that at the time. (If you are of the belief that the nation-tans knew of the concentration camps immediately and did not buy the overwhelming propaganda of the time – which is not the accepted headcanon of the mods –, then Roderich’s disillusionment simply came quicker.)
Resistance groups sprang up once Austrians realized that Hitler scorned their traditions and planned to overwhelm them with totalitarian rule, effectively assimilating Austrians into Germany and ending the Austrian identity (Luza 8). So yes, later Roderich would have grown disillusioned, as the Austrian people did. But initially, the Anschluss would have seemed wonderful and he would have likely been grateful that Ludwig and Germany were going to help him and his people.
—The source cited throughout this is the book Mod 2 is slogging through for a book review: The Resistance in Austria, 1938-1945 by Radomir V. Luza
TL;DR: Roderich was lonely and wanted out of the wheelchair.