Amazon refusing to show me the price of something if I do not add it to cart

That's an odd definition of non-public if the information is available to everyone. More like annoyingly public.

That's a weird logic. If I cannot find it on price comparison sites, the offer doesn't exist.

This is super common with niche hobby products I buy. Doesn’t make any fucking sense. Vendors will send out an email saying “hey we have a sale but we can’t tell you the dollar amount just the percentage until you put it into your cart.” I think it’s also common with some lines of luxury goods. You’ll find a few different reasons online if you Google “luxury brands hide price.”