Looks like the intent com.wireguard.android.action.SET_TUNNEL_UP does not
do much. I tried to start the tunnel like this:

am broadcast -a com.wireguard.android.action.SET_TUNNEL_UP -e tunnel wg0 -n
com.wireguard.android/.model.TunnelManager

After further investigation, the code to handle this intent is deactivated
[1]. Any help would be appreciated!

[1]:
https://git.zx2c4.com/wireguard-android/tree/app/src/main/java/com/wireguard/android/model/TunnelManager.java#n275