This website does have different CSS for mobile devices - is it really the case that just the edit button is missing but the reply and peek buttons are present? If so I have no explanation, but if the other buttons are missing too then I wonder if it's a width problem and the buttons are being displayed but off screen to the right. But in that case the text of messages should run off the right hand side, too.
I have noticed that on a mobile device a tapped button disappears (which is only apparent briefly before the action takes place, usually a webpage fetch), but I think I know why. On a non-mobile device when the mouse hovers over a button it displays a different version of itself that is brighter. But on a mobile device you can only tap a button, not hover over it. Tapping and hovering are like a single step, so the button replaces itself with the brighter version just as the webpage is being fetched. The particular timing of events results in the regular button disappearing before the brighter version replaces it, and then the brighter version is never fetched because the browser knows a new webpage is coming.