While being able to re-run an automation using the current configuration would be quite useful, usually I just re-trigger the automation. When the trigger is a webhook, you can just call the webhook again.
If calling the webhook from the original source is difficult, you can use a script to call the webhook. Since you have a script, you can also have the script console.log the webhook post data in JSON format to make it easy to reuse the original data when you call the webhook again.