Payment method doesn't get saved for free trial subscriptions

[ricardo]

Describe the bug

The issue was initially being investigated in woocommerce/woocommerce-gateway-stripe#1964, but we found it's not coming from the gateway.

To reproduce

1. Ensure you have WooCommerce Subscriptions and Stripe installed and activated.
2. Sign up for a stripe.com account
3. Enable test mode and add test API private and secret keys to your account under WooCommerce > Settings > Payments > Stripe.
4. Navigate to Products > Add new and type a product name: e.g "Free trial subscription".
5. Set the product data to Simple subscription.
6. Add a subscription price and a free trial period.
7. Save the product.
8. As a shopper, add the newly created free trial subscription to the cart and proceed to the checkout block.
9. Complete the purchase with a Stripe test card: 4242424242424242 and any valid CVC and expiration date.
10. Navigate to WooCommerce > Subscriptions and notice the newly created subscription is requires a manual renewal.

The above only happens with the checkout block.

Expected behavior

Free trial subscriptions shouldn't require manual renewals.

Screenshots