You'd have to be more specific about your Kodi needs, but look into 
Shield can easily be controlled via HA's Android TV Remote integration as well as the Android Debug Bridge
What you are describing is possible!
Another good thread
Update: You can launch any app by its package name.
You can launch Kodi with market://launch?id=org.xbmc.kodi Starting with Home Assistant 2024.6 you don't need to add market://launch?id=
Try this in Dev Tools and see if it opens your Kodi App on your Shield device.
action: media_player.play_media
data:
media:
media_content_type: app
media_content_id: org.xbmc.kodi
target:
entity_id: media_player.pc_shield
This is easily added to an Automation based on a specific time trigger. The remainder of the Kodi commands can be applied via the Kodi integration or consolidated to a single script/automation