And the problem only happens when the gateway is set to recurring only.
When it is set to single payments the button appears.
Have reviewed your install and this is because none of your plans allow for recurring payments (see "Auto-recurring renewals (*):" under the Pricing tab of your plans). Setting "Enable Stripe E-Payments" to "Enable single payments" or "Enable single and auto-recurring payments at user's choice" both works on your site. Using "Enable auto-recurring payments" of course does not since you've no plans configured to allow recurring payments.