You should not need to use the Turn On / Turn Off script; those are useful for powering up receivers and switching inputs prior to playback.
The SpotifyPlus can only transfer playback to the official Spotify players (Mobile, Web, Desktop) if they are currently up and running. The integration cannot start the Spotify App on those types of devices (phone, browser, computer). Your Google Cast devices are different, in that they have the Google Cast protocol running all of the time; the SpotifyPlus Integration can launch the Spotify app on those devices, then transfer playback to them.
You can use the player_transfer_playback service to transfer playback to the device, or you can use the player_media_play_context \ player_media_play_tracks services to start play directly to the device of choice. Remember, if it's a phone then the Spotify App must be started prior to the service call.
You can test the following examples via the HA Developer Tools \ Actions tool (using Go To YAML Mode) ...
The following assumes that Spotify is up and running on the iPhone:
service: spotifyplus.player_transfer_playback
data:
entity_id: media_player.spotifyplus_todd_l
device_id: "Todds iPhone"
The following will activate the Google Nest Audio device:
service: spotifyplus.player_transfer_playback
data:
entity_id: media_player.spotifyplus_todd_l
device_id: "Nest Audio 01"