I cannot see this happening in the future. What you're looking at is something similar to visuial voicemail, a service which you have to pay for everywhere else. This is a "no frills" provider, and you only get the basics (calling, texting, voicemail and caller ID).
I think if this option was available I would acutally use VM but at this point and for the last number of years I don't even use VM. And yeah, no way this is going to happen. Other providers use it as an added service. You could see if there is an app that would do it but I don't think so since it would need to have access to your voicemail which is offered through your provider.
@Nicholas255 have you looked into 3rd party services for this? YouMail [web, Android, iPhone] is popular, though I'm sure there's other options out there. EDIT: I think there's either ads or a subscription cost, but you could use some of the $ you're already saving by being a Public Mobile customer and allocate it toward a service like this.
Great idea, i do get that service with my home ip phone service(Nettalkduo) i get my messages straight into my email, it's pratical when your out of the country, i don't see it hard to implement, but hey what do i know?