Mr. Biden’s trip may ultimately do little to alter his political fortunes back home, where his polling numbers are low and he seems headed toward a tight race against his predecessor, Donald J. Trump. House Republicans are itching to push ahead on a long-shot impeachment inquiry, and his son, Hunter, is facing federal indictment on a gun possession charge.

The president is also facing persistent questions about his age and effectiveness as a messenger for his own agenda — part of a broader conversation in Washington about leaders, some with health problems, who remain in office well into their 80s or 90s.

As aging politicians of both parties, including Senator Mitch McConnell of Kentucky and Senator Dianne Feinstein of California, suffer from health complications in the public eye, some are beginning to call for more transparency into the medical history of elected leaders.