What are the benefits of adding the Product Hunt button to your website on launch day?

From a supporter standpoint, it brings a sense of joy and trust when I see the badge/button. I perceive the product to be reputable in solving a real-world problem or bringing something unique to the table that others haven't within their respective markets.

Reputation. Authority. It's an attempt to build credibility

If your website drives good traffic, those people can potentially support during launch. Second, it definitely builds trust.