Like with most scandals with legs, a large part of the Zimmerman scandal in the US wasn't the details of the initial scandal, but how badly it was handled afterwards.
As this question points out, it seems like it would have been more sensible to deny the memo publicly, or perhaps publicly fire the author, disavow it, and apologize. That's what good diplomats would do. And in fact, that's what their lower-level professionals and allies in the US press immediately set out to do.
At first, some sectors of the US papers, especially those of the
Hearst press empire, questioned whether the telegram was a forgery
made by British intelligence in an attempt to persuade the US
government to enter the war on Britain's side. This opinion was
reinforced by German and Mexican diplomats, as well as pro-German and
pacifist opinion-formers in the United States.
Unfortunately for Germany, the country was run by Kaiser Wilhelm, who had the typical personality of an authoritarian politician. He was obsessed with his status, could never be wrong, and had the motivational instincts of a bully. Anyone who objected to the country running its domestic or foreign policies along those lines got sacked, starting with Bismarck (arguably the greatest European politician of his age).
So what they ultimately did instead of any of those sensible professional diplomatic moves, was to deny that what they did was any kind of problem whatsoever, and accuse US officials (and the public that popularly supported them) of being babies about it.
However, on 29 March 1917, Zimmermann gave a speech to the Reichstag
confirming the text of the telegram and so put an end to all
speculation as to its authenticity. By that time a number of US ships
had been torpedoed with heavy loss of life.
...
He also said that despite the submarine offensive, he had hoped that
the USA would remain neutral. His instructions (to the Mexican
government) were only to be carried out after the US declared war, and
he believed his instructions to be "absolutely loyal as regards the
US". In fact, he blamed President Wilson for breaking off relations
with Germany "with extraordinary roughness" after the telegram was
received
This is of course the exact kind of delusional egotiscial bullying that had landed them at war with nearly all of Europe on multiple fronts at once. Denying reality may sometimes work to steamroll local political opposition, but when it hits outside reality the results typically aren't pretty.
