A member of Congress can be arrested when leaving the house to go to dinner, but not when leaving the house to go to the House, or to the Senate.
Does the Constitution make an express distinction between public and private conduct? Yes (the Supreme Court has found private performance of state functions by contract to be state action, but that doesn't remove the distinction). May private conduct interfere with the public interest? No.
Article I, Section 6:
“They shall in all Cases, except Treason, Felony and Breach of the Peace,
be privileged from Arrest during their Attendance at the Session of their respective Houses,
and in going to and returning from the same;”
A member of Congress can be arrested when leaving the house to go to dinner, but (with some exceptions) not when leaving the house to go to the House, or to the Senate. Their work for us can’t be interfered with even to enforce the law against them. Enforcing the law is a public interest itself, but it’s a higher public priority to ensure that the people are represented when decisions are made.
A vital public interest outweighs private and even lesser public interests.
The public interest is sometimes called "state interest". A compelling state interest is a legal standard in court.
But arresting a member earlier or later in the day isn’t a problem. Their House will have time to adjust the voting schedule if feasible to wait for them to address their private emergency (making bail). A clear distinction is made between public and private conduct; and no private conduct, like going out to dinner, can interfere with public conduct, like an arrest.
The Constitution takes the distinction between public and private conduct, and the priority of public conduct, so seriously that commuting to and from Congress is expressly distinguished from other daily travel.