do you have another lamp, fan or something you can test the Kasa switch on? I also found on some devices, if they have any type of digital on/off switch, there can be issues with turning them on.
for example, I have a floor fan with remote control, it will not come on with alexa, but like your lamp, it will turn off. you must be able to leave the fan on in order for alexa to work but when it powers off with alexa, it automatically turns the fan switch off too which makes it useless for alexa. there is a chance something similar is happening with your lamp.