Redstone circuit that activates from two inputs, but only requires one input to stay on

Let's re-frame the problem a bit. What you're really looking for is a garbage disposer, which has a button for turning it on, and an auto off feature when it's disposed of all the items. A T-Flip Flop is not the best thing for this, as it has only a single input.

I would use use a RS-Nor Latch in this case. You can keep the reset line high whenever the chest is empty (simple not gate), meaning that pressing the on button won't even do anything. When the chest has any items, the reset line will be low, and sending the set line high (by pressing the button) will trigger the latch output to go high, initiating the garbage disposal process.

The advantage of this is that this can be made fairly compact. Typical implementations of RS-Nor Latches are smaller than T-Flip Flops, which themselves can be smaller than piston based D-Latches (repeater D-Latches are still smaller than both, but IMHO aren't appropriate here).