Discussion on: Does every backend need to be an API?

Well, every frontend needs storage and a way to interact with that storage. That said:

• if you only have a SPA, then you need to ask yourself what are your storage requirements? Does local storage (with the risk of disappearing) satisfy those requirements?
• an API is an interface to call some functions. What's customarily called REST API is more of a standard to interact with a web service that in turn deals with some sort of persistence.
• if you've already settled on postgreSQL for storage, then probably yes, unless you go for MVC you'll probably end up with an API of sorts.
• MVC is the all in one pattern (think any MVC framework, rails, laravel, symfony, etc - you only have one project). What you mention in the first paragraph sounds more like MVVM, like Angular frontend (or react or whatever).
• also, not all webservices come with APIs, not all APIs need to be REST (the vast majority aren't, they can be RESTful at most, or REST-like since strictly abiding by REST is way too much overhead)
• CRUDs never cover all the interactions with data. It's a fairly pointless distinction that's rarely useful in and all by itself.

I believe that @zakwillis is correct.
In addition I would add that regardless how you interact with your data, the result will be an API. It may not manage a webservice, but will still be an API.