Cthulhu stories and the CoC tabletop game were usually about hapless rubes being caught up in the gears of the mythos. Delta Green operatives are abnormally aware of and equipped to deal with the Cthulhu mythos. It avoids the situations where a private eye, housewife, and baker are thrown up against Nyarlathotep.
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Lovecraft stories make for crummy gameplay. Yes, the characters in the stories are typically unsuspecting buffoons. Do you really want to play unsuspecting buffons? Really? Delta Green operatives know things about the Cthulhu mythos, but they still don't understand (nor can they ever understand) the big picture. Delta Green also isn't a government agency. It was once a government agency and was dissolved to create Majestic-12 (a group of terrible dummies who want to be pals with Cthulhu). The people who work in Delta Green are operating completely in violation of local and federal laws wherever they go. They don't have federal funding, don't go through special training programs, etc. They are just smart, organized people who are focused on defeating the various armies of Cthulhu, Nyarlathotep, the Karotechia, the Fate, Majestic-12, etc.
Personally, I think that works a lot better than playing in a CoC setting and having to act incredibly shocked at every revelation that is familiar to every Lovecraft fan. "THERE ARE MONSTERS WHAT WHAT?! PERHAPS I CAN DEFEAT THEM WITH MY .22 PISTOL AND WALKING STICK! OH NO I AM BEING TURNED INTO A SERVANT OF GLAAKI GURGLE GURGLE."
