God Help Me-
Blame Harry Turtledove for how his reworking of WWI in TL-191 fascinated me enough to read John Keegan and others about the denouement of European liberalism in the totally avoidable brutal crucible of stupidity and agony of World War I.
As to the Zimmermann Telegram, it proved to be the worst brain-fart in a series of German foreign-policy screw-ups, except possibly violating Belgium's neutrality.
However, let's look at why they did so beyond assuming the stupidflu (even more virulent than the "Spanish"/Ft. Riley flu) took out Germany's leadership.
On paper, the US in 1917 wasn't much of a military threat. The US had a decent fleet, but no army worthy of playing on the killing fields of the Somme in numbers, guns or savvy and wouldn't for a year after declaring war in 1917 IOTL between American refusal to learn from the British and French and just training and gearing them up from scratch.
So from the German perspective, the US Navy wasn't going to make the blockade any worse and the US Army was laughably small- 100K all told with obsolete equipment and hadn't exactly covered themselves in glory since the Spanish-American War. They'd get butchered to a man in a slow couple of weeks in Northern France.
Seeing Black Jack Pershing blunder about after Pancho Villa probably convinced a few in the OKH that the Mexicans, if they invaded in force, could tie the US in knots for years. The Mexicans had zero chance of actually defeating the US before getting curb-stomped, but the Germans didn't care. The Mexicans' job was to keep the US from being much of a military factor in the European killing fields.
The Germans'd already spiked the Russians' guns by sending Lenin to raise havoc on the Russian home front who ended up collapsing the Provisional Government and suing for peace with Brest-Litovsk which was a spectacular success. All kinds of unintended consequences to that, but it undoubtedly worked.
Why NOT see what the Mexicans could do IF the US was "just about" to join the fray anyway?
IMO, based on what I've heard American public opinion was decidedly mixed about entering the war. Staying neutral proved quite popular and profitable.
Some folks were understandably upset losing loved ones on Allied ships during USW but they were warned of the risks.
Unless deliberately poked, FWIK the US wasn't joining the fray, especially with the ghastly casualties as the war ground on.
The problem was, as many have said, the US was already doing quite well economically keeping the Allies in the fight with loans and food and goodies as it was. They didn't need to send a single doughboy to defeat Germany in WWI if they actively sided with the Allies. The Mexicans invading wouldn't have changed that picture at all.
So long post shorter, the Germans picked the wrong tool, strategy, and roused the US to action and sealed their doom, based on some very mistaken assumptions.
One, that the Mexicans were so proud of their nation that they wanted to redress all grievances with the US post-haste, no matter the consequences.
Two-that America's resources would have little impact on the war, neutral or Allied.
Three- That America couldn't mobilize quickly enough to be a formidable military power.