1. MC doesn't compare himself to his allies.
2. Chief would not feel outdated after all he was surprised but not disappointed to see Spartan III's on Reach.
3. You hear cortana say "Let's show these guys what a real Spartan can do" but MC didn't say a word and did his job, so he does not feel superior.
4. Also the Spartan IV's have no reason to feel cocky to somebody they know full well could kick their asses without even trying, also Chief did save them when they bumped into Lasky.
English
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I wasn't aware Chief ever stumbled upon a Spartan-III.
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Could of swore he did, unless he was surprised as seeing Spartans that have survived, and it was Halsey who was supposed to see the Spartan III's.
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Halsey thought Noble Team members may have been Spartan-II's but she probably figured they were actually Spartan-III's when she had met other Spartan-III's later on Onyx.
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She knew about the program since before meeting Noble team since one of her AI's (Cortana's older sister) found files in Ackersons files about another Spartan generation.