Oh, this is definitely ~のではない (だろう) か right here, so that part is definitely where the speaker show what he thought or suspected! I don’t know about ないかと though.
Japanese and its weird double-negative and indirect ways of speaking. I definitely remember there was a lesson in N3 Soumatome reading book that contain this phrase (and several more).
I know the verb utagau doesn’t appear, but the teacher explained the construction as being equivalent to meaning “suspect” which is what made me interested in it 
I got this sentence from an English textbook for teaching idioms, so I guess it’s meant to be an idiomatic translation.
I am pretty sure のではないか is the grammar point のではないだろうか (JLPT N2) | Bunpro though, but it is similar to じゃないか (JLPT N4) | Bunpro
I have a lot of trouble with embedded questions it seems, and I’m not exactly sure what is stopping me from wrapping my head around them…
This website looks very useful, thank you!!
How do you get your head around these? They keep tripping me up 
Echoing @additionalramen’s comment about relying too heavily on English – it’s important to remember that different languages are, well, different languages. They independently emerged as ways for people to communicate information by making sounds. They are not sets of unbreakable rules, nor are they beholden to each other. Sometimes grammar patterns can be changed a bit, and some things in one language just won’t make intuitive sense in another.
Functionally for your questions, this means context can change meaning in one language in different ways than it might change meaning in another language, and any particular translation is not “correct” so much as it’s an option.
Something like ではないか is a composite of grammatical chunks that carry some meaning in Japanese, which is more or less explained in the のではないだろうか grammar point you linked.
So rather than worry about what specific word to use, I’d recommend trying to grasp that a speaker is expressing doubt with ではないか, and then thinking about how you might express that doubt in context in English only if you wanted to translate it. A lot of the time you don’t have to translate, though. If you can read/hear Japanese and understand the sentiment, there’s no need to get hung up on what particular English words you would use if you were to translate it. You wouldn’t be constantly translating into English if you were having a Japanese conversation, for instance.
I usually wouldn’t worry so much about the English translation, but in this case I was just really interested in how it could come about that “suspect” was an appropriate translation, especially since I wouldnt have guessed that from the much softer wording on the grammar point I linked I usually read without worrying about these things, but sometimes I come across stuff that piques my interest. I am also interested in translation and interpretation, so I guess you could consider that I am coming at this sentence from that perspective.
I’m now realising I didnt express my intentions/motivations very well so I’m sorry if I confused people 
In English we do have similar structures but they sound much stiffer. For example,
This makes a poor translation choice as the tone is too stiff and the sentence would become convoluted once the と思う is factored into the translation as well, but it is perhaps closer in terms of literal interpretation.
ohhhh I can see how that might correspond (nuance-wise) to the feeling of suspecting something. Thank you, that helps quite a bit. Do you feel that the nuance of “suspect” is a correct application here or not really?
I’d say the nuance of suspect is completely context-dependent on the topic of crime.
私はその店がもう閉まっていたのではないかと思う
If we change it to being about thinking the shop is already closed, you could say that you “suspect it” but it would sound pretty weird, “I think that” would be a much more natural translation.
As @durutticolumn says, it will be context dependent, but I’d say “suspect” is a good translation with no other context. It captures the vibe and is natural English, which is goal of most translations.
It’s been mentioned already in this thread but translation is more of an art than a science. If you’re interested in different translation philosophies and approaches then I’d recommend a book called 翻訳の授業 東京大学最終講義 by 山本史郎. It does a good job of explaining the thought process behind different sorts of translation styles with plenty of real examples between English and Japanese.
Thank you for the book recommendation!!