The detonation produces higher pressure and temperature, causing higher exhaust gas velocities. More energy in the exhaust means more push from the same amount of propellant. About 15% more. The other yet to be realized in practice advantage is that by eliminating the compressor stages on a rocket engine its construction is simplified, and despite needing a much stronger combustion chamber, it could actually be lighter.

If it can live up to more efficient, cheaper, and lighter then it will certainly see widespread use.

There are bound to be issues with new larger designs, but nothing show stopping is expected.

RDEs are more efficient than regular rocket engines. Also, a big part of engineering rocket combustion chambers is balancing power with combustion instability, which can cause flame-out at best, and rapid unscheduled disassembly at worse. RDEs lean into that high energy stochiometric reaction to eke out more efficiency. Due to the tyranny of the rocket equation, even a modest increase in efficiency can greatly increase payload to orbit.

Pushing on solid wall is more effective than pushing on air. Shockwaves are effectively momentarily solid air that you could push on. RDE basically uses shockwaves as turbine blades.

(technically I should say "incompressible vs compressible" but you get the imagery)

I assume because you can use (mini)nuclear explosions in deep space?