Can the shop page show which products ship free?

Yes. Display → Shop Pages → In product lists puts a small Ships free label under the price of every product whose own price reaches your free shipping threshold, in the shop, categories, tags and search (lists loaded with a Load more button or a filter included), in a block theme's Product Collection blocks, and in WooCommerce's older product grid blocks such as Hand-picked Products and On Sale Products. The price is counted as the cart counts it, so adding that product alone unlocks free shipping. Reword it, or empty it to hide it, under Bar → Wording → Label in product lists. It is on for new installs; a shop that had the plugin before sees it suggested once on the Dashboard while the bar is live.