Presidents equate to CEO's. and their role is primarily on the business side of the NFL to ensure financial/ business success for the Owner of said team. Some teams like Dallas Own/Act as CEO/President, Jerry Owns, Son is CEO/Prez.
For the longest time Bears were Owner and paper pusher with zero football sense Mr. Phillips as President, he did a great job of making Bears lots of money, however, on field product suffered many times over, now with Kevin Warren, not only does he have Biz experience in NFL in same position, like in Minnesota, he is a football person as well.
The good CEO/Presidents allow the guy they hire at GM to run the football operations without hinderance and get called upon when asked for finances of said players, FA, extensions, etc, but not hey Kevin who do you like at pick 10?