Mr. Cornyn canceled his speech — his office said he had done so before the Uvalde massacre, after discovering a scheduling conflict — and declared on the Senate floor, “I’m not interested in making a political statement. I’m not interested in the same old tired talking points. I’m actually interested in what we can do to make the terrible events that occurred in Uvalde less likely in the future.”

Senator Mitch McConnell, Republican of Kentucky and the minority leader, has tasked Mr. Cornyn, a close ally, to keep tabs on high-stakes bipartisan talks on gun legislation that can win at least 60 Senate votes to break through a filibuster.