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.
While the Journals clause and the Statements and Accounts clause are the only places
where the Constitution addresses self-governance information directly,
the Preamble implies a bright-line rule, a check on private power.
The Preamble (proposed for restoration to full legal status, see Restoring the Preamble):
"We the People of the United States,
in Order to form a more perfect Union,
insure domestic Tranquillity,
provide for the common defence,
promote the general Welfare,
and secure the Blessings of Liberty to ourselves and our Posterity,
do ordain and establish this Constitution for the United States of America."
Elections are public self-governance conduct, the way we ordain and establish government. Voting is public conduct. Political speech, running for office, or associating for a march or rally are public conduct. Their free and aggregate nature can be seen. Private transaction is private conduct.
Private conduct may not interfere with the public interest,
especially a vital public interest like self-governance (see Public vs Private Conduct).
The Preamble’s implied bright-line rule is neither vague nor difficult to enforce:
Public conduct must be free from influence by private transaction
whatever its intent.
The public conduct of ordaining and establishing United States government by election, and the public conduct of governing the direction of the exercise of United States offices, must be solely financed by the United States.
Amendment XXVIII, (proposed), Section 1:
"The people of the United States
will ordain and establish United States government under this Constitution,
and the United States will fund and certify self-governance during a period set by Congress,
uniform throughout the States,
preceding any vote in an election, appointment or Act."
Amendment XXVIII, (proposed), Section 5:
“The United States will fund the direction of exercise of its offices.”
Implied rules are real. The Supreme Court’s authority to interpret the Constitution is implied.
So, to restore the Preamble’s full legal force (see Restoring the Preamble) would technically invalidate private campaign financing - but expressing and detailing the rule will work better than implying it.