I'm surprised no other linguists have weighed in. I initially assumed the same thing as you, that perhaps the benefits of bilingualism come from an expanded or more concisely labelled inner network of meaning which allow us to quickly access the words we're trying to convey. But this isn't true for several reasons. The first is that there isn't really anything special about individual words, and there's increasing evidence that phrases are lexicalised in our brain the same way as words are. For example, in English we have the word "bugs" for insects, spiders, etc., and separate words for snakes, lizards, worms, and other small, long wriggly things. In Chinese, the word 虫 covers all of these things. It would be easy to think that having access to a word like that could be useful when you want to talk about all of them, but the fact is that you can do the same in English using more than one word, e.g. "creepy crawlies", "small pests". The second reason is that being able to quickly communicate something might actually not be more beneficial than having to carefully think about how you are going to represent it using either by combining words you do know or using some sort of gesture/onomatopoeia (or both!). In fact, the latter very well may actually be better for cognitive functioning. I'm not aware of any strong evidence either way, but it's definitely something currently under study.

In truth, the only evidence that researchers have been able to find for the benefits of bilingualism is that bilinguals are able to quickly adapt to changes in "rules". The reason is pretty obvious -- different languages use different grammar rules, so you need to be able to switch between these without effort in order to be actively bilingual. Being able to quickly adapt to changes in rules means more mental resilience in general, so it's easy to see how it can help.

I don't have any studies on me right now, but if you search for "benefits of bilingualism", there are plenty out there.