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I mean, its not like Chief is completely and totally unquestionably loyal to the UNSC. In Halo 3 he puts his trust in Cortana over Lord Hood's orders + the whole Del Rio plotline in 4. (I would use Halo 5 as an example but I don't like that game's story)
"Master Chief and Doomguy would be best friends." Master Chief was an abducted child supersolider created to crush human rebellions who is unerringly loyal to the fascist military that created him. Doomguy canonically beat his commanding officer to death for ordering him to shoot civilians. No they would not.