Can a product page say that the item gets free shipping?

Yes, once you switch it on: it is off until you do. Switch on On product pages under Display → Shop Pages. Under the bar, the page then says "Add this to your cart and shipping is free." when adding that product, at the quantity chosen, would reach the threshold, and "This item does not count towards free shipping." for a product in a shipping class you left out of the total. Both lines can be reworded, or emptied to hide them, under Bar → Wording. With both emptied, the product page shows the bar alone and does none of the work of pricing the product.