I'm not sure if this is a 1:1 correlation.  If you look at the typical American diet of the farmers from the 1920-50s, these people were extremely healthy and lived long.  Their diet was based on milk, meat, fruits and veggies.  This diet is high in nutrition, good protein and healthy animal fats.  

The main problem in our modern diet is sugar- both processed and from grains (which is not that good for you) and bad fats - which come from plants instead of animals.  If you cut out sugar, grains, and plant fats, this would solve most health problems.

Spurious Correlations.

That's a misnomer. That site is really illustrating that correlation does not necessarily imply causation, not that correlations are "spurious."

Overall, vegetarians have better digestive/skin health than meat eaters, but that's a generalization.  You're comparing 1% of the population against the 99%, of which 90% are extremely unhealthy.  If you compared the healthiest vegetarians to the healthiest meat eaters, there probably wouldn't be too much of a difference.  Most americans eat too much protein (and waaaay too much food overall), therefore their digestive systems are way out of whack because their intestines are too full.  But you can't say that the reason is due to eating meat.  You also can't discount the insane levels of sugar that most people eat too, which slows digestion, causes cancer and speeds up the aging process.

Talk to any vegetarian and they MUST supplement with essential amino acids that can't get in their diet and are only available in healthy red meats and chicken.  A vegetarian simply cannot be healthy without animal fats/products.  (Only exception:  if a vegetarian is able to eat an abundance of fresh, quality fish, this can substitute for meat/chicken, but it's near impossible to find healty, nonfarmed fish nowadays).  The japanese culture used to be the healthiest in the world because they ate quality sushi mainly.  But their foods habits have changed due to communistic laws which reduce fishing and increase prices.

The main problem in our modern diet is sugar…If you cut out sugar…this would solve most health problems.

But not Type 2 diabetes, which afflicts ⅓ of the population; it's caused by excessive fat intake, not excessive sugar intake.

That the meat-mortality correlation is real, see:
Meat intake and mortality: a prospective study of over half a million people

one of the largest medical studies ever performed.

Most scientific studies seem to show that eating meat reduces life expectancy. However, studies also show that abstaining from meat generally reduces testosterone. Many monastic orders mandate a vegetarian regimen in order to reduce concupiscence (which is generally increased by testosterone). It also seems to have a positive impact on life expectancy as the following article talks about: 
http://greece.greekreporter.com/2012/03/15/higher-life-expectancy-associated-with-greek-monastic-diet/