Ah, so goes the game of telephone tag!
Yes, of course a better resume gives you a leg up in your interviews. If you are inherently a better candidate and have better experiences, and if your resume looks better to the interviewer, of course you are on a better footing!
When people say “same footing”, they mean there is no nepotism/favouritism. They mean that, referrals don't help you in the interview stage. They mean, you need to individually prove to each specific interviewer that you are fit, and that happens in a closed room without outside influence.
What else is in that room? Your resume! So, of course a better resume helps you do better in the interview!