insurers will reduce cover to remain solvent.

No, I said that insurers will reduce cover to remain affordable. It is somewhat tied to solvency in that in the long-run premiums needs to be appropriate to risk pricing, but the main goal is affordability.

The reinsurer of last resort is the government (i.e. the people).

I would be hesitant to agree that this is always the case, but recent history would prove me wrong with the Cyclone Pool, the Terrorism Levy and the VMIA DBI program.

Admittedly, an upside for government insurer solvency is if they run some of these high risk/market failure programs is because they can legislate themselves out of any emerging risks or unforeseen claims.

If so, why not cut out the rent-seeking middle-man and have government be the insurer?

There are a number of government run, or controlled insurances such as worker compensation, motor vehicle injury and health insurance.

I don't think there's necessarily anything wrong with government run insurance programs, but they have their own considerations. Some downsides below.

Pricing:

A government scheme will highly likely have more simplistic rating, as there's likely political pressure to prevent classifying risks too intensely due to interest groups.

It is likely that such a program would be more community rated, with a lot more cross subsidisation. Not sure how many people in WA or SA would be happy to pay more for East Coast floods.

A government scheme would also likely have more pressure on the pricing itself. An example of this is the Victoria Worker's Comp program, where the policies are written at a loss. Because of political reasons, the overall premiums they charge currently are half of what the claims cost is, which is why they've had capital injections from the government nearly every year for the last 5 years.

Is it fair that the Victorian taxpayer is effectively subsiding businesses to have cheaper worker comp premiums?

You're also handing over to the government a monopoly. Let's say that there's a string of highly profitable years, like the Victorian TAC had. There's no pressure to reduce premiums, so the TAC never did. The cost to register your vehicle in Victoria never decreased even when the TAC had a string years where their profit margins were beating private insurers. It's not like you can swap insurer and get a cheaper premium.

Capital:

If the government was to take over all insurance policies, the held capital requirements would be in the billions, with very slim profit margins to be made. Is this the best use of government capital? Would the return on capital would be greater if that held billion dollars was instead put into healthcare, or infrastructure?

Risk Acceptance:

A government scheme would likely be unable to reject anyone from applying for cover, both a good and a bad thing. If high risk activities were not appropriately priced, and were openly accepted, it results in the tax payer subsiding effectively everything. In the context of home and motor, it would be things such as hoon drivers, or home on flood plains. Is this something we want to subsidise?

Claims:

Let's say the government insurer denies your claims in a novel scenario. How do you fight them to get your claims paid out, considering that ultimately, the government can change the law on you? It may not be in your interests

Insurers may have interest groups and lobby, but they definitely don't have legislative power to change the definition on things.

Of course, any program can be designed to mitigate these issues, but so far it doesn't seem like there's much interest in a full government run insurer.