Actually we can have a single member LLC where the manager is not the individual. Regardless of whether the manager of the LLC is an individual member or an outside party the LLC does not create unlimited liability exposure for the manager.
The purpose of the LLC is to create limited liability exposure for the manager. One of the main problems that people had with limited partnerships was the unlimited exposure when serving as a general partner. It was this limit on liability that really helped LLCs take off.
If we are dealing with a case of gross negligence, fraud, etc. then no entity is going to provide protection for that party.
I agree that insurance is an important component when we are looking at harms arising from slip and falls, etc. However, the main problem that people are starting to experience with insurance is when we have environmental claims that will almost always be excluded from coverage. In such instances the LLC is the only way a property owner would be able to shelter them selves from personal liability exposure.