The problem here isn't what Cruz could have done. I mean, Senators have a lot of informal influence — there aren't that many people who can call a governor or other state officials directly and expect to get through, you know, but Senators surely can — and that could be put to good use even if there's noting 'official' that Cruz could do. For instance, Cruz could have gotten the FEMA generators distributed and implemented merely by calling around and getting the right people in FEMA in contact with the right people in Texas. But that isn't what bothers people.
What bothers people is the overt fact that Cruz found himself in the same unpleasant situation as the rest of Texas, but rather than responding as part of the community to face the problem — as one expects of a supposed public servant — he acted like a spoiled aristocrat, using his power and wealth to fly himself and his family off to warmer climes while the peons were left to suffer through as best they can. The act comes off as entitled and arrogant; as though he's too important to be bothered by the petty troubles of the citizens who elected him to office. It was a selfish, elitist move, one which wasn't helped by his half-baked excuses and apparent indifference to criticism.
People have a tendency to minimize this as mere 'optics', but 'optics' is a huge part of political leadership: setting the tone and tenor of a political moment. The tone Cruz set here was an air of glib indifference, with the message that each Texan is on his own, and political leaders don't really cares what happens as long as they don't have to be in the middle of it. Unless that was his intention, he failed in his political role.