Running a subscription service with just direct calls to Stripe or similar payment gateways is possible but also needs dedicated person(s) for decent amount of development and maintenance.
Plus features like updating card details, invoice history - all these can be built. Again, more dev work and resources.
Use of subscription platform like Chargebee or Recurly is definitely a great help here.
Chargebee offered a simple pay-as-you-go transparent pricing and almost trivial signup process.