If you think it is hard to get adult participation without food, try having a kids' meeting with no snacks.
1. Principals are great people. They make or break the entertainment value of your PTO. But I would not personally leave any files with the Principal. What you want is an audit, not in the full accounting sense, but just a responsible party who will look over your file and state that you have complied with all of your bylaws and regulations. You filed a Form 990 on time, and can prove it. You have the minutes, with none missing. You have all the bank statements, monthly reconciliations, monthly financial statements as presented to the Board, etc. You won't have to worry about future accusations if you get your files reviewed immediately after the turnover.
2. It isn't clear what you want the treasurer to do. I've been a treasurer. This is not a policy chair. The Treasurer is there to make sure each request for payment is part o the budget and has an attached original invoice. The Treasurer makes sure that cash is handled with some internal controls and that checks are co-signed. Someone else makes policy and decides what are appropriate disbursements that year.
3. I doubt if your Bylaws will let you continue, but even if the Bylaws are that poor, don't do it. Let someone else have your position and you take a new position, perhaps VP. In one of two years, return to your President position if you choose.
As an aside, your statement that 1/2 the kids are receiving proper care is scary. Call your local United Way to determine how that organization awards its grants. Then apply for funding so that you can give proper care to the other half. In general, UW will want you to have a strategic plan. They will want competent people on your Board, and for this Board to enforce the action items on the strategic plan. All employees will need be accountable to the Board; this includes the Principal if that is a paid position. The employees or the center itself will need to receive credible certifications from national or local umbrella groups. Lastly, UW will want to accomplish something, such as Bright from the Start, and you will need metrics in your strategic plan that show your center will use the UW funds to achieve some of the UW objectives in your community. Know that UW will have objectives for various age groups, demographics, opportunity zones, etc. You need only do some of these UW objectives very, very well to obtain needed funded.
