I have not done Whole 30, but did South Beach - way more liberal than Whole 30 - with great success. I estimate I was eating 75-100 grams of carbs per day back then (2003-2005) but it was enough to (presumably) lower my insulin and allow me to lose weight easily (17.5 lbs in 3 months). It also gave me a foothold into low carb where I (re)discovered Atkins and other versions of LCHF (which allowed me to lose additional weight).

Whole 30 may not be as rigorous as how you eat, but it's a giant leap from the SAD diet. We do not see potatoes as particularly nutritious, but I don't think they are anti-nutritious like grains. I have no idea your daughter's weight or how much she seeks to lose but I doubt potatoes are going to spell doom for her. She's losing weight now - eating potatoes, presumably - why fix what ain't broke?

One other factor, and I'm trying to say this as gently as possible.... Girls/women have enough problems with body image without their fathers (who, face it, many girls adore) telling them how to eat. So she's not eating as perfectly as you? Do not let the perfect be the enemy of the good. Quietly enjoy your daughter taking responsibility for good nutrition and let her figure out what is best for her.