The financial aid adviser didn’t exactly speak the truth: private colleges DO NOT use retirement savings and private colleges will use different formulas. Saving is always preferable to not saving and is used much less than income is. Some colleges use home equity but some have a cap, others don’t, and some colleges don’t use equity at all.
Run the NPC on several LACs of interest, a couple reaches, a couple matches, a couple safeties. Bring the numbers to your parents. See what they say.