A single member LLC provides some benefits but those benefits often require a lawyer to invoke (i.e. you're getting sued, gotta file things and work the legal system). If you have insurance you just tell the insurance company and they hire the lawyers.
Then there's a technical issue at launch and orders are being rejected left and right.
They could be out basically the whole value of the marketing campaign which is 10x your salary. You might owe them compensation for that, unless of course you got paid through an LLC and only have the $100 in your account.
(ps. With some careful lawyer drafting to exempt things which can't be limited in that way (negligence etc.), while carefully wording the exemption so any part voided doesn't take the whole limitation with it.)
On the other hand various types of corporate insurance cover you for particular risks. Depending where and how you operate you may have to have them by statute or by practicality. So you may have to carry liability insurance by law if you have an office where people visit, or a contract may require that you hold E&O insurance up to a certain amount. In a way the latter is actually your customer protecting themselves from your use of a LLC. Without it, in the case of a settlement against you, you could easily just turn around and say "fine, the corporate account has $5 in it, here you go" and then fold up the company leaving them with no recourse. With E&O coverage for certain types of errors, they know they can get covered in a settlement up to a certain amount.
E&O covers you for particular failures in providing the service you are contract for. Say you were an electrician and did some work on a new building. They sue you claiming your work wasn't to code and caused them $100k in trouble with the city - you disagree. E&O insurance covers your court costs and potentially your settlement if it goes that way. It isn't going to cover you if you get sued for libel because of things you said about them, etc.
This is also why, for example, it may be hard/impossible to get underwriters for some software consulting. Because there are not professional standards groups that are well recognized and because potential damages from software can be difficult to asses (your script change cost us $10mm in AWS fees) insurance companies may not want anything to do with it.
So that leads to a third prong of protections which you didn't mention, which is you need to pay attention to your contract terms (and set them as much as you can). Depending on your situation this can range from easy to impossible, but can have a huge impact. For example, I've successfully added clauses to limit all liability to actual spend on previous 6 mo.
[1] this is hugely dependent on jurisdiction, particularly with single member variants.
But it's a one-time fee, so you're probably still right in general, but for the moment I'm mostly coasting on savings & sponsorships, focusing on the stuff I care about while I have the breathing room. I might consider being a bit more organized if/when I ramp up business again and am considering looking for new clients.