Can shoppers check delivery to their pincode from the cart page?

Yes. The cart summary has a box that says Enter pincode for delivery with a Check button. When the customer types their pincode and clicks Check:

• a green tick and your serviceable message appear if you deliver there, or a red cross and your not-serviceable message if you don't
• COD not available here is added when Cash on Delivery is switched off for that pincode
• the Shipping line and total refresh for that pincode

The messages come from your Delivery Restrictions settings. Without that plugin, the box still shows the shipping estimate.