I believe that there are very few schools that offer the deals that Harvard etc. do to low income families. Perhaps just the Ivies and Stanford. It is accurate that those few schools will offer such good deals.
However, very very very very few kids get into those schools. And your kid has to have a lot more than grades to get in. They have to have excellent grades, phenomenal test scores, plus be state-level awesome in sports or music or something, do community service...and then their chances are about 10% after all that.
And then to get those good deals, you have to be both income and asset poor. Furthermore, there are two financial aid systems in the US - FAFSA and CSS. The asset hiding you're describing helps with FAFSA, it will not help as much if at all with CSS. Guess which one the Ivies, Stanford, and most of the best liberal arts schools use? Yup, CSS. (Well, they use both - FAFSA for government aid such as Pell Grants, CSS for institutional aid.)
I would suggest sitting down with your kids and making a list of the schools or kinds of schools they might apply to and get into, then look at those schools' financial aid situations. You can get a very good estimate of what aid you might receive by using the schools' estimated cost calculators - each school is required to have one.
Two more things: As economista points out, the school is not required to meet full aid. So if the school costs $50K a year and your EFC is $0, the school may give you $25K in aid and say the rest is up to you. Secondly, even if a school meets 100% of your aid, they can "meet" that aid with loans on the student and/or parent(s). Rarely if ever does a school meet need with all grants/scholarships.
That all being said, there are things you can do to improve your chances of aid, particularly with FAFSA schools. The web has a zillion resources on this. The last tidbit I would point out is that FAFSA is now a prior-prior year system, so if your kid is starting college in the fall of 2023, then the first income tax return that you need to optimize is calendar year 2021. Be aware and plan ahead.