The Zimmermann Telegram was a crucial component in the United States' decision to enter WWI. However, had it not been intercepted and Mexico had received the message, would Mexico's political and economic climate have allowed them to attack the States, and was the government willing to do so given the promise of German support?
I just want to make something clear as the wording of your post makes me think there’s a misunderstanding regarding the telegram. Yes the telegram was intercepted, but that doesn’t mean the Mexican government didn’t receive it - they absolutely did receive it, and so it was, in fact, an open offer that they were prompted to consider (obviously the Mexican government chose not to accept the offer for numerous reasons, which other people have in this thread have provided more information on).
To address the possible misunderstanding though, I’d like to clarify something. Intercepting a telegram doesn’t function like intercepting, say, a mailed letter does. Obviously, if you intercept a letter, you get the information from it but you also stop it from reaching its destination, but that’s not inherently the case with intercepting a telegram (although completely blocking the message could still be done, if desired, but severing the wire). The signal can simply be copied as it passes through some point in the wire, allowing it to be intercepted and read, without disrupting or blocking it. At the time, the British were doing this with all traffic that passed through the Porthcurno relay station on the westernmost tip of England, which included the Zimmerman telegram.
This comment has a good insight on the Zimmermann Telegram, although this comment's reply might be a better answer to your question.
But, in short, Mexico was dealing with a bigger problem themselves, as the Mexican revolution was responsible for the death of many mexicans and the country was struggling internally. As such, it seems unreasonable that Mexico would engage in a war with the US.