Remove `woocommerce/subscriptions-core` from `composer.json`

[mattallan]

Project Thread: pgizk6-42-p2
Related Epic: #10401

This issue is being handled by the Chronos team as part of our Subscriptions Primary Offering project, as such, I've added this to our team's project board.

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• Targeted release: WooPayments 9.4 (May 21st, 2025)

Description

There are a small number of US merchants running legacy implementation of our Stripe Billing subscriptions feature which is built on outdated/deprecated subscriptions code. Back in 2023, we attempted to upgrade these merchants over to using the WooCommerce Subscriptions.

This issue is the final part of a 6 part series which ends with us removing the deprecated woocommerce/subscriptions-core composer package from WooPayments. Those small number of US merchants still using the Stripe Billing feature without WooCommerce Subscriptions installed, will need to install and activate WooCommerce Subscriptions to continue selling subscriptions on their store.

Acceptance Criteria

• Remove woocommerce/subscriptions-core from composer.json
• Clean up any subscriptions-core code found in WooPayments
  • Deprecate wcpay_init_subscriptions_core()
  • Deprecate is_wcpay_subscriptions_enabled()
  • Deprecate is_wcpay_subscriptions_eligible()
  • Update should_use_stripe_billing() to not return true when is_wcpay_subscriptions_enabled() is enabled but the subscriptions extension is not enabled.