Season seven of Halo: The Master Chief Collection is set to address this and so far, I'm impressed with what I've seen based on preview 'flight' versions of the game sent out to 343's dedicated Halo fans.
The Anniversary Edition upgrades were all present and correct, but 'classic' Halo was built from Gearbox's 2003 Windows port, which doesn't hold a candle to Bungie's Xbox original. We first took a look at Halo Combat Evolved on PC over a year ago, concluding that while it was a good enough port of the existing Xbox version, the foundations of the port themselves were flawed.