What is the difference? The reason why it is alarming is because there is no political separation. It doesn't really matter which way it goes.

This happened in the 70s. Nixon politicized the role with Burns (who came from the CEA), then Carter appointed Miller (who went onto become Treasury Secretary). And btw, Miller was (with hindsight) one of the worst Fed chairs of all-time.

Tbf, Volcker came from Treasury, Geithner went into Treasury (after FRBNY) but, in both cases, they operated with Presidents who respected the distinction in functions. This distinction weakened significantly under Trump, and is now non-existent.

Is the big deal primarily that the Fed needs to be able to spike rates (like Volcker) to get rid of inflation and close a business cycle, but that tanks asset prices and forces Zombie businesses to finally die, which is politically toxic, so it doesn't happen if there is too much political control of the Fed?

Correct. In 1957 or 58 (I can't remember which), the Fed increased interest rates to remove excess out of the economy. Economy went into a fairly mild recession. The Fed gets blamed for causing the recession (the Fed Chair at the time said the Fed should to take away the punch bowl...that stopped happening). Nixon loses to JFK (remember he was Eisenhower's VP, so Nixon blamed the Fed when he lost). And the cycle that led to the inflation of the 70s (where monetary and fiscal policy is timed to the election cycle) begins, tacit political involvement).

Also, before 1951 (and for a period of years after, although the formal break was 1951), the Fed wasn't functionally independent from govt. Because the war debt was so large, the Treasury used the Fed to press interest rates down so the debt could be paid down (it continued after 1951 because the debt was still really huge, note the similarity with your hypothetical). So the period at the end of the 50s was the first real test of Fed independence.

Again, I don't think people today understand that Fed independence is clear legally but has been more flexible in practice. Why? Because setting interest rates is inherently political. And there is an asymmetry: the incentive is always to be loose. The lesson is that there is no real way to get around political control, because the temptation is too great. I also think that policymakers should rely more heavily on macroprudential policy to take the heat out of markets (this is happening in the UK) because normal monetary policy is so asymmetric.

How do you see things playing out now that we're near 0 interest rates and in some cases negative? Will governments continue to be loose via negative interest rates, through some other mechanism, or will they be forced to tighten policy?

In theory as you approach zero smaller changes produce bigger results in an aysmptotic fashion, but in reality lenders have margin to think about.

Maybe the zero rates will take into account lenders margin, and once that is taken into effect the rates will reflect the asymptotic curve.