No, I really think that you do need to be dispassionate about what you keep and what you cut. You need to forget your investment and look at it as someone else who doesn't share your investment in the work. You need to give them what they need, not what you want to include.
Aristotle (Poetics) said "Characters are reveled through action." Think of Back-story as two functions: It helps the writer create whole characters that will be revealed through their actions, and it is an Antecedent Event such as writing the murder of John Doe so that you know how Dead John got in trouble with the killer and what clues the killer left, as well as the how and why. Once you have this Antecedent Event down, you can send your sleuth out to find out what happened: Who did it, Why it was done and how it was done. The Sleuth picks up clues and eventually find out this material, as does the reader. The problem of Back-story growing into The Story is solved by understanding why you are writing backs-story for these characters. If you understand why, you can easily choose how much.
Your mistake is assuming by story I mean 'plot'. I should hope not. I use 'story' in the most inclusive sense of everything that makes a narrative worthwhile: the journeys of its characters, the texture of its world, the moral worldview it explores, all these things that are interconnected and which serve a central but expansive narrative idea. Anyone who thinks everything should serve the development of the core plotline is a hack, and I'd be insulted to be thought of that way.
I'm with @minstrel . 10K Likes to @jannert ert for this! Please, someone, put @jannert 's above words into the Writing Advice Hall of Fame. @jannert , you are a treasure.
A lot of people will say there's some sort of formula, or are emphatic about how much to include or not to include, but I really think it depends on the story. So, only you can know it at the beginning as you're writing it, and then you have to rely on feedback from your Alphas and betas to tell you what's germane and what needs to go. The three fiction projects I have in the pipeline (current, future #1, future #2) require different amounts of backstory for the main characters. Current WIP is about a composer who defected from Russia, who'd been recruited into the Soviet system as a child prodigy. He's been living away from home since age 12, was completely on his own since age 14 when his last loved relative died, and he escaped Russia by himself at age 18. All that stuff affects the way he relates to the current world around him in New York, the level of trust he has in other people, and even how he loves other people, so more of it is incorporated into the story. Next WIP, Future WIP#1, is about a rock star on the road as he and his band become famous. His backstory doesn't matter so much, because the story is about the wild ride of fame. Next WIP, Future #2, is about a female rock star who is connected to this guy ^^^. The story takes place during a period of time when there were very few female musicians on the road and when there were only a handful of female rock stars in the entire music business. It was a time when women absolutely could not have husbands and kids and be rock stars. So, knowing some of what makes this chick tick and why she's made the choices she made will be important...yet the backstory will still not as important as the Russian guy's in my current WIP. So what suits your story? What immediately informs that person's decisions and actions and how that person relates to other people? I would say, at this point, you know your characters pretty damn well. So tell their story, man. Stop procrastinating and write the story!
Like the others said, add the backstory only if it works in your story. You can also have your characters talk about their backstories when they are in a conversation. And they can think about a backstory when something comes up. Keep it as a short paragraph and stick to the main plot.
On a practical note—and sticking to one of my favourite sayings about writing: Write without fear; edit without mercy—when you're writing, write everything that comes to you that you want to write. However, after a suitable cooling-off period, where you have 'forgotten' what it was like to actually write the piece, then go through it and decide what should be kept and what should go. If you come to a scene or chapter that you have a niggly doubt about, ask yourself, "if I get rid of this part altogether, what will happen to the story? Will it screw up the plot? Will it lessen the reader's immersion in the story's world? Will it make what the characters do seem too abrupt, or unnatural?" If the answer is 'nothing' will change, then I suppose it could go. Cut it (but save it) and again, let the changes sit for awhile. (Make sure you also deal with any references to what you've cut.) You will probably find when you go back to read the whole thing again much later on, that you don't miss this bit at all. Cutting it was a good idea. If you find that you do miss it, however, you can restore it whole, or modify it. I've done both myself. Cut and forgotten, and cut and later replaced. It's all down to how you feel on the day, versus how you feel later on. There can be a difference between what you think about as you're writing a story, and what you see as you're reading it with fresh eyes. It's a matter of distiguishing between what you meant to write, and what you actually ended up with. I think it's important to recognise this duality phenomenon. It's as if you become two different people. It takes distance from the story to achieve this duality. And good story editing takes more than one or two pass-overs. Checking for SPAG errors doesn't even count in this process. That's proofreading. This is editing. If you don't want to pay a professional to do it for you, you've got to learn to do it yourself. Like anything else about writing, it's part of learning the craft. You'll start out making mistakes—not seeing plot holes, or gaps that need to be filled, or piles of stuff you don't need after all—but you'll get more proficient at it, the more you do. Just don't rush the process in the zeal to 'get published.' Sketchily-edited work won't get picked up by a traditional publisher, and it will embarrass you later on, if you self-publish too soon.
I think @deadrats is right, in that a writer should be passionate. However, I agree with LostThePlot that an EDITOR should be dispassionate. If you're going to wear both hats, you need to be able to shift from passionate to analytical when it comes to editing your writing. Passion gets the initial story out there, but it can blind you to flaws. Editing ensures that the writer's passion will transfer to the reader without glitches or misunderstandings. Both are creative tasks, but require different skills.
In this day and age how many people can toss off a first draft and leave it to an editor to figure the rest of it out? We're all editors to some degree, we have to be to get anyone else interested. And, honestly, even if we weren't obliged to submit well polished, proofed, intelligible manuscripts I think you'd still be well advised to do so. No-one writes in perfectly polished prose. Even those who can write very tightly to word limits or to carved in stone plans of the content still need to weed out their typos and make sure they used their semi-colons correctly. Writing and editing are different skills certainly, but they are different sides of the same coin. If you can write but can't edit you won't get published. If you can edit but can't write you work as an editor not a writer. You do need passion but you also need to temper that. If your process runs that way it's ok to just write whatever takes your fancy. That's certainly what I do, or what I used to anyway. But slowly that was tempered with realism and a better understanding of what other people will be OK with. I love me a downer ending, I love bleak soul crushing ironies, I love playing scruples through the narrative and I get a kick out of trying to make the audience figure out whether they are ok with what they just read. But I don't really do that any more. Ever new project has been more mainstream, dialed back the qualities that made them almost totally unmarketable. I still have passion for my projects, I still take a great deal of joy from them. But that comes with a dispassionate understanding that you just can't write a book that's quite that dark if you ever want to see it in print. Passion and dispassionate are Ying and yang. You need both and you need both all the time. The passion to make something feel yours, and the distance to make yours something that others will be passionate about too.