Thank you for you suggestion! This is really great feedback. Currently the "insert social media icon" option is available in the Event Invitation but not on the Event Landing Page and Registration Form. Please keep voting for this and other features you want added to EventSpot. We're always looking for ways to improve this product.
Also, since we were talking about social media and in case you haven't tried it yet, you can Simple Share your entire event and not just the invitation. Just click Simple Share on your event's dashboard.