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Note that Noether's second theorem still applies, so any timelike vector field will yield a conservation law due to diffeomorphism invariance. In that sense, energy conservation still holds even in the absence of Killing fields, though some people argue this doesn't count as a 'proper' law of energy conservation as it includes contributions by the gravitational field, for which an energy-momentum tensor cannot be defined...

I'll say that most systems are time invariant, so conserve energy. One should always know the system one works with. For example, first year undergrad problems near the surface of the earth for example since they ignore the earth do not conserve momentum.

"Noether's" theorem. Is that a joke or what?

Nope, named for Emmy Noether.[0]

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

Wow, and in 1915, I doubt a scientist never had a more fortuitous and timely name :)