OP: I think you need to stop thinking of 10 gauge as some magical ideal. It is not. It’s just better by immeasurably small amounts per foot and most 120V outlets will accept it.
At 10 Amps of draw, the hypothetical 30’ 10 gauge wire would drop 0.6V vs. the 0.2V of the 6’ power cord. Tiny amounts relative to 120V incoming AC.