That claim made by Nikki Haley probably killed what little chance she had of winning the nomination. Consider the downside...
Older voters are increasing as a percent of the population, and they vote in greater numbers than other age groups.
Why should 75+ be the standard rather than have all candidates be required to test? Many 75 year olds have served with distinction while, for example, Pennsylvania just elected a Senator (not yet 75) who is suffering from the effects of a stroke!
And if anyone thinks that universal absentee ballots and drop boxes etc, which have recently been introduced into our voting processes are controversial, subject to tampering, or the like, consider what requiring a medical test in order to run for office would do.
Who gets to choose the doctor? How many doctors should independently concur? What if there are conflicting opinions? How often should the test be given? What if there is a medical question, but the condition is temporary? Are the doctors free from political bias? And who's to say that THIS PART of the many obligations that candidates already need to fulfill (petition-gathering, reporting the financial backing, residency requirements, for example), all of which have been "fudged" by some people before, will not be repeated with this requirement?