General relativity has no problem with dark energy or dark matter. Singularities, yes. And nonrenormalizability is an issue (with trying to make it quantum).
That's right. The problem is more that it doesn't say anything useful about dark energy and dark matter. Apart from allowing us to infer their existence.
General relativity doesn't say anything useful about electromagnetism or baryonic matter either; I'm not sure you can lay an absence of an explanation of dark matter at its feet. For dark energy of course is much harder to say...
It feels natural to expect it from General relativity as the dominant force at cosmic scales is gravity. The various MOND proposals are admittedly niche, and we might eventually be able to fully explain Dark matter without such modifications.