Why could it be argued that Progressivism was a conservative movement?
I rather doubt that it can be considered progressive. The entire program of the Progressive movement was reform. The mantra of the movement was that the cure for the ills of democracy are more democracy. Although two Presidents, Roosevelt and Taft, might be considered progressive, they were not typically conservative. Roosevelt went after big business with a vengeance in his "trust busting" campaign. His activities upset J.P. Morgan so much that when Roosevelt went big game hunting in Africa, Morgan commented, "let every lion do his duty."
Big business was often the target of the movement, not its darling. At a time when a high tariff would have supported big business, President Taft worked for a lower tariff. The progressive movement also supported the nomination of candidates for political office by primary. The old system, nomination by convention, allowed party insiders to control matters. Additionally, the 17th Amendment was passed which provided for direct election of Senators.
I'm not sure where the idea came from that the Progressive movement was conservative; but anyone who argues that position, in my humble opinion, is fighting a losing battle.
The Progressives were very conservative (in some ways) when it came to social issues. That is how you should argue this. Progressives wanted everyone to behave in a certain way -- they wanted especially to reform the working class and immigrants so that those people would act more like middle class "natives."
The clearest example of this was the movement for temperance and Prohibition. This is a very conservative idea -- that people should act in very sober and "moral" ways. Conservatives tend to like to have the government intervene in people's lives to force them to act in "moral" ways and this was a very good example of that.
So, insofar as social issues are concerned, the Progressives were pretty conservative because they wanted to enforce their view of morality on other people.