Can I show recommendations on products where I have not picked any related items?

Yes. Use Default products (Optional) in Plugins → Up / Cross Sell:

• Upsell tab: these products show on any product page that has no upsell products of its own.
• Cross-sell tab: these show in the cart when none of the cart's products have cross-sell products set.

Search and add products in Search products to add…, then click Save Upsell or Save Cross-sell. Products where you set your own picks keep showing those.