In fact, a jet engine could in theory (ignoring things like heat rejection) operate just fine in space if you brought along a massive bag of air and hooked it up to the intake duct. But then you've basically just designed the world's worst rocket engine :)
Using atmospheric oxygen for the early phase of flight is a massive weight savings. If/when SABRE works, the Skylon spaceplane could be a practical reusable SSTO. It could carry a very respectable payload mass (the aim is 11-17 metric tonnes) to LEO, and it could do so quite cheaply if the designers' claims pan out.
Fun fact for Kerbal fans: the RAPIER engine in the game is inspired by SABRE.
Either way, an LED is not generating any meaningful thrust, and pointing that thrust at a sail adds no benefit.
A propeller makes thrust like a paddle, by pushing off of something. A jet/rocket engine makes thrust by recoil, using conservation of momentum.
But even if they did not have fans, there is no fundamental difference between using a propeller or fan to speed up the flow of air and heating the air inside a chamber to speed up the flow of air. (The fan is used because at subsonic speeds it can be made more efficient.)
In other words, all 3 means of propulsion work by conservation of momentum.
There is a whole separate device called an EM Drive, which claims to basically violate conservation of momentum. It is highly controversial, and unlikely to be anything other than a measurement mistake.