I still do not have the 74HC08 AND chips, so I moved ahead and used what I have. I took out the 00 and replaced it with a 74HC02 NOR chip. NOR is not or. OR logic yields a HIGH result (or TRUE, or ON, or 1) when either input is HIGH, and LOW if both are LOW (or FALSE, or LOW, or 0). NOR is the opposite--only HIGH if both inputs are LOW.
I was reminded that NOR chips are the only ones with different pinouts, so I had to do some minor rewiring, but that's OK. Once I got the chip in, I pushed the buttons like yesterday. On power up the LED is on, just like yesterday. However, if either or both buttons are pushed, it turns off.
Next I wired up the NAND chip again, moving the LED and 1K resistor down to its output, connecting one input to power (so it's always HIGH), and the other input to the output of the NOR chip. Now the light will turn off if the output of the NOR is HIGH. We know how to do that: as in the previous paragraph, if neither button is pushed, it sends a HIGH signal to the NAND chip, making both inputs HIGH and turning off the LED. If I push either button, the signal is LOW, and NAND turns on the LED. Very cool.
Here's the video.