Materials, Reagents, and the Cradle are all fine. Do keep in mind synthesizing materials will take some time.

However:

[quote]~The Lesser Grail of the 3rd war was destroyed, preventing victory from being claimed. This is why the Church were called in as mediators[/quote]

[hider=Church][img=http://i.imgur.com/4hLQlN5.png][/hider]

They were called in after the Second War because everyone was like "WHOO NO RULES NO PARENTS". 

Third War became a battle over the Lesser Grail with the Nazis and whatnot, but I don't recall it ever being said to have been destroyed.

[quote]~Gilles de Rais, a notably evil man, is summoned into the 4th war as a 'Heroic Spirit'. This shouldn't have been possible with the Grail's system[/quote]

That wasn't really public knowledge. Even the mathematicians aren't going to know something like that.

[quote]Despite being in a position to win the war, Kiritsugu destroyed the 4th war's Lesser Grail[/quote]

See above.

[quote]The Lesser Grail has been destroyed before without interrupting the 60 year cycle, so the theory that the mana from the 4th war was recycled being the cause of the premature 5th war doesn't hold. [/quote]

Nani? 

[i]Q: Even though it should only happen every 60 years, was the 10 year gap between the Fourth and Fifth Holy Grail Wars as short as it was because the Lesser Grail was destroyed before the Greater Grail could direct the accumulated mana to establish its pathway trapped some unused mana in the Grail?

A: That’s true. In fact, you could say that the function of the Greater Grail was already muddled when when it was impregnated with Avenger in the Third War.[/i]

[quote]Similarly there was no widespread destruction following the 3rd war's Lesser Grail's destruction; unlike the disaster that followed the destruction of the 4th war's Lesser Grail. [/quote]

Can you give me a source on the Third War's Lesser Grail getting destroyed? It was a fight over the Grail, but like I said, I don't recall it being said to have gotten destroyed.

[quote]Investigation of Kiritsugu past (while difficult, 6 years chasing leads is a long time) could turn up a number of freelance sealing designations and his behavior during them. With that data the recurring ideal of 'sacrifice the few for the many' could be seen, bringing into question his reasons for destroying a wish-granting grail that could carry out his ideal.[/quote]

See the Kiritsugu thing above.

[quote]From my point of view, the bare minimum conclusions that can be drawn from a thorough investiagtion are that the grail has been tainted by something literally evil, it no longer grants wishes, and the contents are dangerous. While she might not know it as 'mud' per se, and certainly hasn't made the connection to Angra Mainyu, she understands that the contents are in an altered state and pose a danger to the world.[/quote]

Like I said, the suspicion that something's "wrong" with the Grail System is fine, but it'd pretty largely have to be vague. The level of information you want just isn't available.