Mr. Page and Mr. Brin have largely avoided public events, with Mr. Page having a medical ailment that has made public speaking difficult and Mr. Brin focusing on the company’s more experimental projects. Mr. Pichai has tended to speak mostly about products and services, instead of policy.

Through a Google spokesman, Mr. Schmidt declined to comment.

Mr. Schmidt has been marginalized over time at Alphabet through a combination of its changing leadership, the shifting political environment in the United States, and his own personal gaffes, according to people familiar with the company, who spoke on the condition of anonymity because they were not authorized to speak publicly.

Since 2011, Mr. Schmidt has been a go-between for the company in Washington — some people internally referred to his role as Google’s secretary of state. During President Barack Obama’s administration, Mr. Schmidt, who has supported many Democratic politicians, prominently represented Google on policy matters. He also gave money and technical assistance to Hillary Clinton’s campaign team during the 2016 presidential race.

But since President Trump came to office, Mr. Schmidt’s standing has changed. He has been eclipsed in Washington by others at Google, including Susan Molinari, a former Republican Congresswoman from New York, said some of the people familiar with the company. Google also has new Washington staff members such as Max Pappas, a longtime political operative who has a relationship with Charles and David Koch, the billionaire brothers who support conservative causes. In the meantime, Google is under fire, along with other tech giants, as legislators seek to deal with the perceived monopolies these companies have.

In a time of heightened scrutiny on workplace behavior and sexual harassment, Mr. Schmidt’s personal life has also attracted attention. While he is married, he has brought a series of girlfriends to corporate events over the years. Though Mr. Schmidt’s relationships were outside the office, the fact that they were carried out publicly and that the women attended professional events with him set the tone for other executives, several former Google executives said.

Some gaffes by Mr. Schmidt over the years have also received attention.

In a 2009 interview on CNBC, for instance, Mr. Schmidt said something about Google users’ concerns about privacy that still haunts the company today: “If you have something that you don’t want anyone to know, maybe you shouldn’t be doing it in the first place.” Google has also had to tamp down reports that Mr. Schmidt sought to remove personal information, including about political donations, from the search engine, which he has denied.