I have Starz in my Roku channel and have the Roku mobile app but on my iPhone it only gives the option to play on my TV
I think that's just the way it works. On the Roku app on a mobile device, you can view the channels that come with the basic The Roku Channel, but not the extra subscriptions that you can pay to add to it. Roku only has the rights to show such subscriptions on Roku devices.
If you want to view a premium channel on more than one platform (web browser, one or more streamers, etc.) you generally need to subscribe directly with the channel on its web page, then log into its dedicated app version or web access on each platform.
I think that's just the way it works. On the Roku app on a mobile device, you can view the channels that come with the basic The Roku Channel, but not the extra subscriptions that you can pay to add to it. Roku only has the rights to show such subscriptions on Roku devices.
If you want to view a premium channel on more than one platform (web browser, one or more streamers, etc.) you generally need to subscribe directly with the channel on its web page, then log into its dedicated app version or web access on each platform.