Hugh here from Shopify. I hate to bring bad news to you, however, it is not possible to have the Favicon used on your site show in the checkout page on the store.
This is because the checkout pages are hosted on our secure server, separate from what is used to display the rest of the site, before the checkout pages. All Shopify stores on standard plans experience this, and no custom Favicons can be used. The only plan type that can customize checkout pages are Plus accounts.
Please let me know if you had any other questions about this!
Is shopify planning to add this feature for basic tier shops?
This should be included as a basic feature. Shopify plus is far out of reach for startups.
Do your competitors offer it? Seems so...