I wish I would have seen this thread sooner. Bird is either an English House Sparrow or Tree Sparrow, both exotic to US and not afforded protection by migratory bird act. Based on photograph it fledged too early. Once fledged unless a lot younger than that, the chicks will not stay in the nest. That pattern holds for most songbirds regardless of nest site. In most instances I have seen prematurely fledged English House Sparrows, there was a powerful motivator to exit nest and odds of survival already greatly reduced.
Your efforts to preserve chick are neither misguided nor a waste of time. It helps you develop a familiarity with wildlife increasing odds you make knowledge based decisions on it later.
Now for the butt chewing! Do not feed such birds crap like yogurt or other processed foods. Immature birds similar to yours eat insects almost exclusively. They need to be fed frequently food items that are roughly the size of the eyes of the birds being fed. I raised many small birds including Barn Swallows, Cardinals, Starlings, Sparrows and American Crows. It was a major time investment. Result was me having birds following me around for days even after flying about and starting to feed themselves. If you are going to school or work that interrupts potential feeding schedule from dawn to dusk, then odds of bird thriving are low until it begins feeding itself. That chick needed about another week of such intensive love. The bird was also a couple days shy of being able to thermal regulate well unless in a temperature range of 85 to 95 F. I have used heat lamps but do not like it as well as having a bunch of chicks snuggled together.
More details could be discussed if situation comes up again. My kids are nearing age where they will likely attempt similar although we will most likely try European Starlings. Imprinting and weaning issues will arise for us as they likely would have for you. Long term survival odds low unless kept indoors as adults. Rehabbing efforts by the "pros" do not fair well post release either. The birds need to learn which means the ability to explore the outside world.