How to trigger Alexa routines from Home Assistant

This may be a stupid question but… have you disabled and enabled (and relinked) the skill on the Alexa app? Also, have you run a scan after that?

You can try checking for events. On the Alexa Developer Console, on the “Test” tab, enable the Device Log and the Alexa Smart Home (Beta) widgets. On the Smart Home widget, enable the live debugger. The, on the device log widget you should see events appearing every time you change a sensor or turn on a light that is exposed to alexa.

EDIT: if you still see the LWA auth errors, then my suggestion above will not work. But those errors could be from you testing the lambda directly.