I still think presidents have an advantage in winning shutdown fights with Congress. But Democrats have a bigger advantage over Republicans.
That advantage has three sources. One is the media’s liberalism. A second is the public’s association of Republicans with anti-government sentiment: If someone has shut down the government, the enemies of government are the people you’d expect to be responsible. A third is that in every shutdown fight I have ever seen, some Republicans cannot resist saying that their party should bring on a shutdown, predicting that nobody will notice it, and making other comments that encourage the media and the public to blame them. President Trump has played that role this time.
In this case, there’s another factor: Republicans have the White House and a majority in both houses of Congress. Of course they would get blamed for a shutdown.