Thanks for posting those notes which refute your argument even more. Madison feared election by the legislatures and set the system up to prevent that.
“If the magistrate does not look forward to his re-election to the Executive, he will be pretty sure to keep in view the opportunity of his going into the Legislature itself. He will have little objection then to an extension of power on a theatre where he expects to act a distinguished part; and will be very unwilling to take any step that may endanger his popularity with the Legislature, on his influence over which the figure he is to make will depend. 3. To avoid the third evil, impeachments will be essential, and hence an additional reason agst an election by the Legislature. He considered an election by the people as the best, by the Legislature as the worst, mode.”