The Spartan 3s were made years later and never even set foot on Reach.
Actually, Reach fell in 2552, and Gamma Company (Third Spartan III Company) was active in October 2552 (Ghosts of Onyx). Alpha and Beta, where Noble Team are pieced together from, were used several years before 2552. So the timing for a Spartan III being around isn't in question, it's whether they were on Reach.
Which, Noble Team is a secret team, made up of surviving (and ONI claimed
worthy) Spartan IIIs who survived their suicide missions and put into a team, like Headhunters, except with HH they were taken out of training for more advanced training after a certain set of skills are met.
Kai, Emile, Carter and Jun (Alpha) would be dead from Operation: PROMETHEUS, as would Kat (Beta) in Operation: TORPEDO,
if they were participating. They could have been delayed for some unknown complication or injury, like what would have happened to one of the teams in my fanfiction of the battle. Or simply they could have been transferred to another section of the S-III Program like Jonah and Roland into the Headhunters sub-branch of the S-III Program.
So from my mind, there are too many possiblities for this to actually make sense in arguing if a Spartan III could be on Reach, saying like
The timings wrong! or
There were no S-IIIs on Reach!. The timing for a Spartan-III to be around is correct, and like the Headhunters the team didn't necessarially need to be survivors of the battle, which is already slated in canon as no one surviving PROMETHEUS, so the only logical thing to consider with all the Noble Team members is that they were transferred, like the military always does. They will take an outstanding member of the Navy Seals or ODST and throw them into a group to match their skills, whether it be a black ops team for the USA or a secret sub-ODST division in Halo that we haven't heard about.
So all things considered, the timing is correct, and Noble Team could be just a team of outstanding members of the S-III Program, ONI Section-III Beta-5 making them get transferred to Noble Team for their even more secret stuff that the S-IIs can't be called to do, The UNSC using them to battle the Covenant openly and bring moral to the population.
Timing: Correct
On Reach: Iffy, but partial solution
But can't we agree, even if there are a ton of loopholes in the game if they aren't explained (or what I theorized), that Reach is gonna kickass?