Nothing is broken. An embed is drawn as a still picture until you click it, and clicking loads the real thing in place. That is true in the editor and on the published project page, and it exists so that a project holding a dozen videos does not try to start a dozen players at once.
Click the picture, not a play button
In the editor the still picture is the control. Click it and the embed is replaced by the live version, which then behaves normally: play, pause, scrub, fullscreen. It stays loaded for as long as you are on the page, and reverts to the still picture when you come back.
There is also a View fullscreen button in the row of controls that appears when you hover an embed. That opens the real embed straight away in a viewer over the page, without loading it into the gallery. Use it when you want to check one embed and leave the rest alone.
A tile reading "Click to load" is normal
Some sources give us no still picture to show. When that happens the editor draws a plain tile with a code icon and the words Click to load instead of a thumbnail. It looks like a placeholder for something that failed, and it is not: click it and the embed loads exactly as any other would.
Whether a thumbnail exists is decided by the source, not by you, and no setting changes it. A video link usually carries one. A model, a prototype or a code sample usually does not.
What a genuine failure looks like
A real failure announces itself rather than sitting quietly. On the project page it appears as a warning icon with Failed to load embed content. If you see that instead, the embed is the problem: the source may have been removed, made private, or may refuse to be shown inside another site at all.
Which embeds you can use in a project covers what is accepted, and why a link can be rejected before it is ever added.