Imagine you try the zombie thing and eat a lot of dead flesh. You would get really sick and maybe die.
I don't know the chemical names offhand, but suffice to say the decomposition process has plenty of nasty byproducts, not to mention whatever might have been in the intestinal tract (eating feces == bad).
In all fairness, zombies don't generally eat heavily decomposed bodies. They tend to eat recently living or even currently living flesh most of the time, hence their tendency to kill people for their flesh.
On a slightly more serious note... there are plenty of carrion eating animals. Many of which are capable of eating humans. I'm sure it's not that toxic.
Apologies if my example was too beyond the pale, and as you mentioned, even less effective due to the freshness of the flesh.
iirc the carrion-eating animals have evolved particular mechanisms such that eating dead flesh doesn't harm them; they can eat a lot of stuff that would kill us. What is deadly toxic for a human may be nontoxic for a vulture or cockroach.
I don't know the chemical names offhand, but suffice to say the decomposition process has plenty of nasty byproducts, not to mention whatever might have been in the intestinal tract (eating feces == bad).