So, first, don't worry about punitive damages. They are very rare, very hard to get, and don't apply in the kinds of claims you'd be likely to be involved in -- and if they did, I doubt insurance would cover that anyway. Just behave reasonably and you should be fine. What you have to worry about are things like pain and suffering, economic loss (if someone can't do their job any more), and long-term care costs. Which can be huge in and of themselves.
The way it was described earlier is what I recall: the existence and amount of insurance is discoverable, but it cannot be presented to the jury, because you don't want a jury to say, hey, the guy has a $2M insurance policy, let's award $2M in pain and suffering for a $25K injury, it's just the insurance company. Now, personally, I don't think you should be able to talk about finances the other way, either; the jury's job is supposed to be to figure out who was at fault and how much that injury cost. If the plaintiffs can't point out that the defendants have insurance to cover those costs, then the defendants shouldn't be able to whine about how the claim will bankrupt them.
As a practical matter, yes, most cases settle for the insurance limits. Because most people don't have money! But that's not an actual rule or requirement. If you get a $3M judgment against you, and you have $1M in insurance, the plaintiff has ever right to expect you to pay the $2M difference out of your own assets. Yes, some assets will probably be protected, and that's going to be different in every state, but you have a legal judgment against you for $2M, and they can recover directly from you.
I don't think it's reasonable to expect perfection or insure against every risk, and I'm not trying to. But I got umbrella insurance years ago, because I know I'm a target. I'm a lawyer. People are going to look at me and think I'm rich, even if I'm not -- and I'm going to get zero sympathy from any jury around here. And I don't want one stupid or thoughtless moment (and I have ADD, so I definitely have them) -- to wipe out everything that I've worked for for decades. I think I have $2M. So if I take out Justin Verlander, I'm fucked, but it should be enough to entice a settlement for the vast majority of the kinds of claims that I could be involved in.
And yes, I did need to have certain auto and home limits to qualify for the insurance. But I already had those higher auto limits out of the same concerns that led to the umbrella insurance.