They choose graphite as a moderator because graphite is a pretty good moderator. Graphite also had the advantage of being cheap, inexpensive, and it did not cost too much. When you need many tons of the stuff cheap is good.
Back in the 1950's Graphite moderators were pretty much what you had to work with. The U.S first reactor was graphite moderated.
It is true that Graphite will burn but it won't burn if it is inside a nuclear containment vessel with no oxygen. All is swell until the vessel breaks and lets out the coolant (liquid or gas) and lets in 02.
With the proper design there is nothing wrong with graphite moderated reactors. About 20% of today's reactors are graphite moderated.
"U-238, the most naturally abundant uranium isotope, is, like I said, 3 times as common as silver. It is not fissile, but breeder reactors make it so by bombarding it with neutrons."
Technically correct but incomplete. 238U is non-fissile. There is nothing you can do to 238U to make it sustain a fission reaction. However.....
238U is considered a Fertile Material. Which means you can turn it into a fissile material by smacking it with neutrons. Smack 238U the right way with a neutron and it will turn in to 239Pu (after a short period cross-dressing as Neptunium-239)
239PU and the heavier Pu's (241, 243) are fissile and can be used in Plutonium reactors
God I love nuclear stuff.