The first step in troubleshooting is to make sure you have the "smart" features turned off, namely Home/Away Assist and Eco Mode. (I have only troubleshooting advice. I don't know anything technical about the Nest or how to fix it. )
With the Nest set to 67 and the room temperature at 70, what does the Nest tell you that it's doing? Does the screen acknowledge that it's supposed to be calling for cooling from your AC, or does it show that the room temperature is above the target temperature but it doesn't seem to care, or does it show that it believes the room temperature is already down to 67? If you push the target temperature down to, say, 64, does the Nest respond and cool the house?
