Tim Walz had scarcely set foot in Nevada on Tuesday afternoon — the third of four states he was to visit that day and the beginning of a high-stakes campaign blitz in the American Southwest — when he found himself, once again, in trouble.

That morning, Mr. Walz, the governor of Minnesota and the Democratic nominee for vice president, had charmed donors at private receptions in Seattle and Sacramento with jokes about Minnesota weather, debate prep and Donald J. Trump. But Mr. Walz had also, twice, expressed support for getting rid of the Electoral College, the system of presidential vote apportionment laid out in the U.S. Constitution.

This opinion did not reflect the official position of the Harris-Walz campaign, a Walz spokesman was quick to clarify in the ensuing kerfuffle. But, more broadly, it did reflect the gamble that Vice President Kamala Harris made in choosing Mr. Walz as her running mate: that his unscripted, freewheeling rhetorical style might break some china but might also endear him to the American electorate.

“I have a habit of speaking before I think on some things,” Mr. Walz told an audience of donors in Nevada later on Tuesday afternoon at a private home set in the lush, rolling ranch lands outside Reno.

“Nine weeks ago, I was sitting in my house, minding my own damn business,” he said, to a wave of laughter. “And now I’m here, trying to do my best, at times saying stupid things,” he added, to more laughter. “But I’m trying to make the point: We so deeply believe in this country.”

Mr. Walz made that point of almost breathless optimism, in discursive speeches that defied punctuation, at every stop he made across the Southwest this week, as he pressed the ticket’s case in the critical battleground states of Nevada and Arizona.