> You can kick the problem to a "neutral commission"--but that's giving tremendous power to unelected people

It works pretty well in Australia. Here, federal election district boundaries are drawn by the Australian Electoral Commission (AEC), an independent federal agency. I've never heard anyone remotely mainstream suggest the system is rigged or gerrymandered or politicised. (There are certain elements of "unfairness" built into the system – most notably, that the least populous state gets the same number of Senators as the most populous state, just like in the US – but the AEC didn't decide that, the Constitution says it.)

The AEC has total control over federal elections–the states have no role in decisions regarding them. For state/territory and local elections, each state and self-governing territory has its own electoral commission, similar in principle to the AEC – although some of the details, such as how exactly it is appointed, will differ. Unlike the US, in Australia, local government has zero role in running elections, even local government elections–all control of elections is at the state/territory and federal levels.

The AEC and its staff (and their state/territory equivalents) are required to be politically neutral. They are not allowed to belong to political parties or publicly support or criticise them.

The chief executive of the AEC is appointed by the government for a five year term. Although government politicians (effectively) choose the chief executive, they are required by law and custom to choose someone without a record of past involvement in politics – the current incumbent used to work in a civilian executive role for the federal law enforcement, before that as a manager for a defence contractor, and before that he was in the military – so obviously someone with some demonstrated experience in management/leadership but without any known political affiliation.

As well as its chief executive, the AEC has two other members – a chair and a third member. The chair must be an active or retired federal judge – the Chief Justice of the Federal Court of Australia gives the government a list of three judges, and the government gets to choose one from that list to become the chair. The third member is also chosen by the government, but legally must be head of another federal agency – (non-binding) tradition dictates that the government choose the head of the Australian Bureau of Statistics (ABS), the Australian Statistician. Major decisions–such as drawing electoral boundaries–must be made by the Commission as a whole, by a two-thirds majority vote.

> With a "neutral commission," by contrast, you've created a body with a lot of power that can't be voted out.

Nobody can vote out the AEC, but the government can remove its three members from office for "misbehaviour". However, the removed member can challenge the removal in the courts, so if the government tries to remove one without a strong justification, the courts are likely to strike down the attempt, whereas removing them with strong justification is likely to be upheld. To my knowledge, no removal has ever been attempted. (Also, the AEC is established by statute not constitutionally, so Parliament could amend the law to abolish the AEC and replace it with something else–highly unlikely to happen in the foreseeable future.)

I can't see why you couldn't use a similar model in the US, and why something like that couldn't work there. (You couldn't copy the exact Australian model into the US, due to various differences between their respective national constitutions, but you could translate some of the major ideas of the Australian model into something that could work in the US context.)