Which role of a senator takes place only at the Capitol?

The Senate is responsible for all lawmaking within the United States. The Senate also has important powers under the "advice and consent" provisions. They adjudicate proceedings initiated in the House of Representatives, a two-thirds majority being necessary for conviction. They consider and process legislation and exercise general control over government agencies and departments.

I believe the answer is: serving on a committee

Within the committee, senators from each state have to form relationships with other senators that might have opposing view.

After that, they would debate their view in the chamber and eventually vote for the type of legislation that give the most positive benefit for the country.