If you draw the boundary enclosing the power source, you still have a closed system for which conservation laws ought to work, so long as nothing travels through the boundary. It's not having no source that makes it closed, it's having a boundary that leads to all the sums having to add up.

So traditional thrusters, which also convert energy into momentum, shouldn't work either?

Conservation of momentum only applies to mechanical processes, such as force and collisions. Converting energy breaks out of those bounds, and so it no longer applies - and the emdrive breaks no physics.

At least, that's how I've always understood it.

No, momentum is simply outright conserved in the same way that energy is conserved. There are the two simplest to derive conservation laws, stemming from spatial symmetry (momentum) and time symmetry (energy).

These relationships are derivable via Noether's theorem, one of the most foundational results in physics.

https://en.wikipedia.org/wiki/Noether%27s_theorem

They eject something from the system.

Rather the closed system includes both the craft and the ejected mass.

If you include the ejected mass, then momentum is conserved in that system. The ejected mass goes one way, the rocket goes the other. The momentum of the rocket is equal and opposite to the momentum of the ejected fuel.

Not true for the EmDrive.

Conservation of energy and momentum are separate phenomenon and in this context, "closed" relates to the latter. You can have systems like particle collisions in a collider that receive energy from (are accelerated by) the electromagnets but the collision is a closed system because when the products scatter, they conserve the momentum of the colliding particles.

I think maybe the "system" is only with respect to momentum in this context.

I have a similar question. How can a system be truly closed ? It’s always interacting with the stars and galaxies through their gravitational fields. Similarly, and closer to your question) the spent energy can only be Generated through a chemical or nuclear reaction (both of which should lead to a loss of mass, which will be impacted by the gravitational fields, making it a non-closed system), or by gathering energy from outside The system (solar panels or other technologies), thus making it a non-closed system

What am I missing here ?

It can often be considered closed for the purposes of the phenomenon that we want to describe. To be specific, masses like galaxies, stars and planets accelerate all parts of our tiny rocket the same way. Yes, there are differences, but they are usually so small that they can be neglected. Our rocket works mostly the same no matter how it is oriented to these sources of gravity. Of course, as it gets closer to such masses, these forces (called tidal forces) start to matter and will eventually rip it apart, given that it doesn't burn up or collide first.

Rockets, differently to balloons, do not depend on an interaction with a gravitic field for their operation. They work by ejecting reaction mass opposite to the flight direction and relying on Newton's third law of motion. Most of the time, the energy to eject the reaction mass is generated by combustion, but it doesn't really matter. Rockets can also be driven by the reaction mass's pressure inside its tank.

In general, any system can be turned into a closed system by including all sources of force in the description. Of course, the more forces are included in our model, the less useful it gets. Therefore, physicists and engineers must decide which forces can be ignored. Else, even experiments would become useless because without a model, the experience gained from experiments cannot be generalized.

As to your first question, distant gravitational fields can be ignored as per theory of relativity - if you're looking at an experiment in a black box (e.g. in an EmDrive) then an otherwise closed system in freefall in gravitational field is absolutely indistinguishable from a system at rest without that field, any experiment would give exactly the same results.

And for the second question, mass is energy and energy is mass (as per relativity). Adding or removing energy from a system makes it respectively heavier or lighter; if you heat up an object so that afterwards it has exactly the same atoms but just more internal energy, then it will be slightly heavier than when it was cold - the difference is very small (m=E/c^2, and c^2 is large) but nonzero.

If you're looking from outside at the behavior of an isolated black box, it's impossible to detect if inside it some mass is "lost" in a nuclear reaction i.e. converted to energy, as both the inertial and gravitational mass of a system is equivalent to the sum of both rest mass and energy, and that total is conserved. If that box pushes out energy/mass to outside or gains energy/mass from outside, then of course it is not a closed system. But converting rest mass to energy and vice versa in a chemical or nuclear reaction would not make it a non-closed system as long as all the inputs and outputs stay in the system, as the total gravitational effect of the system before and after the reaction would be exactly the same.

Also heat, they talk about 1kW energy. Some heat has to go away, or the whole thing would glow and melt soon.