Debug a WooCommerce checkout that broke after a payment gateway plugin update

A WooCommerce store's checkout broke immediately after updating a payment gateway
plugin. Here's what's known:

SITE: [WooCommerce 9.x, WooCommerce Stripe Gateway updated from 7.4.0 to 8.0.2 this
morning]
SYMPTOM: [checkout spins on "Processing payment" for 30+ seconds then shows a generic
"error processing your payment, please try again" message]
IMPORTANT DETAIL: [some customers report their card WAS charged, but the order in
WooCommerce still shows as Pending payment]

Walk through a triage in this order, and for each step state what result would confirm
or rule out that cause:

1. Check the WooCommerce > Status > Logs for a fatal-error or gateway-specific log
   entry from today, since a plugin update can introduce a fatal PHP error on the
   checkout template that a generic front-end error message hides.
2. Confirm whether the gateway is still correctly pointed at live mode API keys and not
   accidentally reset to test-mode keys during the update, which is a common cause of
   "payment declined" errors that look like a broken integration.
3. Check whether the webhook endpoint URL or webhook signing secret changed in this
   version, since Stripe (and most gateways) uses webhooks to confirm payment success
   asynchronously. If the signing secret is now mismatched, Stripe can successfully
   charge the customer's card while WooCommerce never receives the confirmation and
   leaves the order stuck on Pending, which is the exact "charged but order says
   pending" symptom described above and the highest-priority thing to rule out first,
   since it risks the customer being charged twice if they retry checkout.
4. Rule out a caching conflict: confirm the checkout page itself isn't being served from
   a page cache that predates the plugin update, and that no full-page cache is caching
   the nonce or cart total.
5. If none of the above resolves it, roll back to the prior gateway plugin version on
   a staging copy of the site and re-test checkout there before doing it on the live
   store, to confirm the update itself is the cause rather than a coincidental timing
   issue.

Replace the bracketed plugin, version numbers, and symptom with your own site's details.