Does investing in the S&P 500 for 3 years yield better yearly returns than investing for 15 years?

It depends which 3 years (and which 15). If you had bought the S&P500 index fund in the 2004-2005 time range, your 3 year returns would be small, nothing, or even negative, depending on how exactly you timed it, whereas waiting 15 year (until about now) would have more than doubled (nearly tripled, again depending on exact timing) your investment. See the below chart from Google:

You mention getting 9% over 3 years, but 7.6% for 15. That only works if some of the other 12 years have smaller returns. If you "do the 3 year hold 5 times" (how exactly is this different from just holding for 15 years, other than potential transaction fees and capital gains taxes every 3 years?), some of those 3 year periods will have smaller returns.

"the average return for 3, 5, 10, 15 years is 9%, 8%, 13% and 7.6% respectively"

I suspect this really means that the returns using the S&P average over the previous 3, 5, 10, 15 years is … . The answer you are quoting from seems to be simply giving examples of how the market has performed recently in order to give an idea of what is a reasonable expectation.

I.e. if you had invested your money 3 years ago you would have made 9%, and if you had invested your money 15 years ago you would have made 7.6%.

It says nothing (and can say nothing) about how the market will perform in the future.

I know that my logic is missing something crucial as I always assumed that the longer you leave your money for the closer it approaches to the long term average return. If you can get 9% a year for 3 years, but 7.6% a year for 15 years couldn't you just do the 3 year hold 5 times?

The error in this logic? How will you know when to buy and hold for only 3 years and then do it successfully 5 different times?

Here's 20 years of S&P 500 performance in percent:

 1998    28.6      
 1999    21.0      
 2000   - 9.1      
 2001   -11.9      
 2002   -22.1      
 2003    28.7      
 2004    10.9      
 2005     4.9      
 2006    15.8      
 2007     5.5      
 2008   -37.0      
 2009    26.5      
 2010    15.1      
 2011     2.1      
 2012    16.0      
 2013    32.4      
 2014    13.7      
 2015     1.4      
 2016    12.0      
 2017    21.8      
 2018   - 4.4    

You are correct. The variance in the returns is due to the data points. Short term periods (3 years) including 2002 and 2008 will be poor, even negative. A 3 year return involving the best years but not including 2002 and 2008 will shine. Timing is everything :->)