[_jal]

The comparison to a lawyer or doctor doesn't work.

Your doctor may have many patients, but they are not making the same decision for all of them. Same with any lawyer you retain.

Your elected rep has to represent people with diametrically opposed desires. Also, you don't pay them directly and you have no principal-agent relationship.

[shuntress]

The comparison does work.

In the analogy, the doctor's patient or the lawyer's client represents the electorate as a whole.