“To govern according to the sense and agreement of the interests of the people is a great and glorious object of governance. This object cannot be obtained but through the medium of popular election, and popular election is a mighty evil.”

Delegacy and deputation at natural law, representing an individual sovereign's inalienable rights in a limited body politic of servants is a great and glorious object of liberty.

Burke fails to mention "which peoples' interest." Banking people, corporate people, etc..

It's a conundrum. That is why the objects of election must respect the boundaries of inherent rights of the individual.

In a representative republic based on nature's law, individuals do NOT ! ! ! vote on their interests BUT RATHER, on what laws (that already exist in nature) to administer and how those laws are to be managed. Interests (by example living off of other's labors and money) that are self aggrandized — such as socialism, are contrary to law and the best interest of humanity. 

The selection process should not concern itself with elected candidates, but concern should focus on elected terminologies, ideologies, ideas and directions of governing, to find the general and best interest of the citizenry.