For term life you absolutely care about the financial stability of the company. Term life isn't "short-term" like 6 months, it's "short term" like 10-20 years. Property insurance is typically a 1 year term and the difference between a company that looks like it will be able to pay it's bills for 1 year versus one that can pay for the next 20 is huge. Just ask pets.com.
As you point you, who know what the future will bring? Volatility is an insurance company's enemy. Take a look at the volatility of the actuary tables over the past 20 years for middle aged adults. And we see almost no change. But we also need to protect the premiums for the market risk of Pet.com from blowing up. However, rates are calculated using 20 year government bonds with no inflation protection, so low risk there. Capital requirements tend to be low because of the low risk. So yes, 10 to 20 years in short term in insurance lingo for such a dull boring product. Well, maybe medium term.
Your understanding of whole life is totally incorrect. Whole life is a life insurance policy that does not terminate after a set number of years; rather, as long as you can cover the cost of premiums, it continues to be in force. An annuity is an entirely different product (although it can also be sold by insurance companies).
Technically you are correct, but I alluded to that. Ask people on what they have got and most will answer whole life. Peak underneath the hood and 9 times out of the 10 you will see that the majority of the payments going towards an annuity.
As for dying in year 29 of the 30 year term policy, he is referring to the fact that since you are still in the term, you should get the death benefit (whether you paid as a single premium or annual premiums is not so important). The problem is if the company goes bankrupt in year, say, 24, then you don't get a death benefit. True, you aren't on the hook for premiums after the company goes belly up, but if you get 30 year term insurance as a healthy 35 year old, then the company goes bankrupt after 24 years (when you are 59), you are in big trouble. You were paying relatively cheap premiums that took into account that you have been paying since you were 35, but now you have to go find another company and get a new policy, now as a 59 year old. And if you have developed health issues since then it's even worse.
Technically you are correct here but reality is different. Rates are calculated using actuarial tables and long dated government bonds – both are low risk. State regulators require reserves and segregated risks. I can't think of the last time an insurance company got into trouble for writing term or whole life insurance. When AIG blew up it did not affect their whole life policy holders because of the safe guards in effect. Once again, a low risk, low capital line of bossiness.
Now, take a look at the average life insurance company. For "Life" products, annuities and the like dominate the balance sheet. Often by a factor of 10. While I can't think of an insurance company that has gotten into trouble over their term or whole life, there have been many companies that have had issues with their annuities and long term care. Actuarial tables have not moved much for middle age individuals – not many die. Figuring out when old people is harder and most of the risk here is that people will live longer, not shorter. This helps the whole life side but that tends to be the smaller side.