Check that the rule is switched on, that its scope actually includes the product (right product, right category, or storewide), and that its schedule dates are not in the past or future. Then run the product through the “Why this price?” tester to see which rules matched.
Open the Rules screen and drag the rules into the priority order you want, then set the combine mode to apply the biggest, apply the first match, or stack them all. The tester shows the result.
The table needs the standard WooCommerce product summary hook. Most themes fire it, including Storefront and Twenty Twenty-Five. If a heavily customized theme removed that hook, the table cannot render there.
No. There are no external API calls and no tracking. Everything runs on your own server.
If something does not work on your store, open a thread on the support forum with your WordPress, WooCommerce, and theme versions, and a note on the rule you set up. We read every thread.