Well, according to the Founders, they are supposed to investigate the candidates and deliberate before voting. This was discussed in the Federalist Papers, No. 68, regarding the function of presidential electors:
"It was equally desirable, that the immediate election should be made by men most capable of analyzing the qualities adapted to the station, and acting under circumstances favorable to deliberation, and to a judicious combination of all the reasons and inducements which were proper to govern their choice. A small number of persons, selected by their fellow-citizens from the general mass, will be most likely to possess the information and discernment requisite to such complicated investigations."
As indicated in Federalist No. 68, the Founders were confident that the electors could pick qualified persons to be president:
"The process of election affords a moral certainty, that the office of President will never fall to the lot of any man who is not in an eminent degree endowed with the requisite qualifications."
Like I said, the only people who want to change the system are people who just don't like a particular choice that the electors have made for reasons having nothing to do with the Constitution. But, it doesn't matter. They are few in number.
Looks like it is the Party's responsibility to verify qualifications here in Texas.