It is true that reaction time increases with age starting in the mid-20s. The response time increases more quickly after the late 60s and onward.
Statistics do show that older drivers (generally defined as over 70 rather than 60) are involved in a disproportionately larger number of accidents than their percentage of the driving public would predict.
As for no one driving after 60, that is just plain wrong (not to mention inflammatory). There are ways for older drivers to improve their driving ability, including making sure their vision and hearing are corrected, not drinking alcohol, making sure medications don't interfere with safe driving, always wearing a set belt, and many other things. One thing that older drivers have that many younger ones don't is experience and maturity.
Here are some sites you might want to check out:
http://www.livestrong.com/article/442800-how-does-aging-affect-reaction-time/ (This one has several articles on aging and driving.)
Remember that statistics are for a huge number of people. They don't necessarily extrapolate to any one individual driver.
Please let me know if you have questions.
All the best,