What are the advantages and disadvantages of a grammatical gender system?

A grammatical gender system (I believe that is the correct term) is when every noun in a language is assigned to a particular gender, often masculine / feminine. Some languages add in a neuter gender, while others have genders unrelated to male/female. Languages that have this feature often require that verbs and adjectives (and other words) change to match the correct gender.

The genders in a grammatical gender system are not assigned to the objects, but are assigned to the words themselves.

English has three gender categories: masculine (he), feminine (she), and neuter (it); however, this is not a grammatical gender since verbs/adjectives do not have to change their spelling/pronunciation to agree with the gender of the noun. Also, the only nouns that are masculine/feminine are nouns that actually refer to something that is male or female.

Grammatical gender systems are incredibly common. They are found everywhere. This makes me wonder if there is an advantage to having one.

As far as I know, the main disadvantage of a grammatical gender system is the time taken by foreigners to independently memorize all of the gender assignments. It is important to point out that this disadvantage does not affect the native speakers of the language, only people who are trying to learn it as a second language.

Also as far as I know, the main advantage of a grammatical gender system is that it reduces ambiguity from pronouns.

However, I know that I must be overlooking something. Surely my above thoughts cannot account for the fact that grammatical gender systems have existed for thousands of years?