Advisers to other prospective Democratic candidates for 2020 acknowledge that Mr. O’Rourke is worthy of their concern. His record-setting success with small donors would test the grass-roots strength of progressives like Senators Elizabeth Warren of Massachusetts and Bernie Sanders of Vermont. His sometimes saccharine call to summon the nation’s better angels would compete with the likely pitch of Senator Cory Booker of New Jersey.

And his appeal to some former Obama advisers — and, potentially, his electoral coalition of young people, women and often infrequent voters — could complicate a possible run for former Vice President Joseph R. Biden Jr., who would aim to win back many of his former boss’s constituencies.

Mr. O’Rourke would surely have vulnerabilities in a primary, including an absence of signature policy feats or a centerpiece issue to date. In his Senate race, he was often disinclined to go negative, frustrating some Democrats who believe he wasted a chance to defeat Mr. Cruz, and he struggled at times in some traditional formats like televised debates. He is, by admission and design, not the political brawler some Democrats might crave against a president they loathe. And his candidacy would not be history-making like Mr. Obama’s nor many of his likely peers’ in the field, in an election when many activists may want a female or nonwhite nominee.