First, those are not flaws that can't be worked around, but it's not likely a character that makes a good protagonist (although it can be a protagonist character, but in a specific genre... more in a moment.). Normally the characters I see are "too many positive characteristics", which is the problem with every single Mary Sue type (although a dark and brooding loner with trust issues are not atypical of Mary Sue, especially if the character is male, so your not out of the woods, yet).
Typically this is a role held by a major supporting character and not the hero, as they tend to be implied to be the guy who does the stuff that needs to be done but the hero might not be able to do... or point out to the hero the flaws in the protagainst's approach to dealing with an antagonist.
That said, your hero would fit right in with the Noir genre, especially as a hard boiled private eye detective stories. This does take some skill to make into a character that is actually someone you want to watch and the story should focus on getting over some of these flaws. Consider the hero of the greatest Noir film ever made, Who Framed Rodger Rabbit? (what, you thought I was going to say something like The Maltese Falcon? No. First, I've only seen parodies or homages of it, not the actual film... which brings me to my next point, in that there are more parodies of Noir style than straight uses of Noir. Rodger Rabbit, for all it's humor, is a straight Noir plot and most of the film's humor comes from the nature of the titular client being a Cartoon, not from mocking the genre.
Our protaganist is Eddie Valiant, a private eye who from the get go is established as the kind of person who, if you heard someone call him a real dick, you'd probably be safe in assuming they meant the term as an insult rather than a comment on his profession (please let me know if I need to explain this joke). From the moment we meet him, it's clear he hates Toons (in this story, cartoons are made like live action films with sets and actors who are played by actors who are ethnically "Toons" and are treated as Stars and Celebrities if they make it big in Hollywood like live action actors.). His first line is a dismissive "Toons," followed by taking a swig of liquor. When discussing the case with R.K. Maroon, the head of the studio that produces Rodger Rabbit's cartoon (one of their biggest sellers, making Rodger an A-Lister like Mickey Mouse or Bugs Bunny), Eddie almost refuses the job of finding evidence of Jessica Rabbit's infidelity because he does not do any work in Toon Town, a neighborhood in L.A. where the toons live, and he only takes it when Maroon informs him that Jessica works at a nightclub outside of Toon Town that, while run by toons, entertains humans only. A few scenes later, Eddie violently assaults a drunk in a bar after he reveals that the word on the Street is Eddie is working for a Toon. The rage comes out of nowhere and startles everybody but the bar owner and long time friend of Eddie explains the source of the rage: A toon killed Eddie's brother (Dropped a piano on his head).
It's important to note that the delivery of this line initially comes off as comedic, this is the only time the death of Eddie's brother could ever ellicit a laugh from audiences, and each time the brother is brought up following this, it's clear that he was Eddie's best friend, his partner when they worked on the L.A.P.D. where they worked the beat in Toon Town, and Eddie's own P.I. buisness is named Valiant & Valiant, implying both men quit the force to go into the business together, where they worked cases for Toon Town's most elite (a montage of mementos from Eddie's career shows that he and his brother at one point cleared Goofy of espionage charges and Eddie owns a Toon Gun that was given to him as a Thank-You gift from Yosemite Sam.).
What's brilliant is that most of the details of Eddie's flaws aren't overtly discussed. Eddie's backstory is shown over a montage of news paper clippings, pictures and awards kept in his office, all of which show him and his brother as a pair or mention them as a duo. The death of the brother is given to us upfront and explicitly given as the reason for his hot temper around the toons. Even the crime that Rodger is framed for (Dropping a safe on the head of another human) is noted as eerily similar to the murder of Eddie's brother. All of this comes to an emotional head when Rodger, the only important person to the film not in the know, after suffering more from Eddie's gruff attitude to working with him and failing to get a single laugh from the man, asks Eddie what happened and for the first time, Eddie actually tells the definitive story. Eddie and his Brother were investigating a Bank robbery in Toon Town and caught a lead to the perpetrator's hide out, only to realize too late that the lead was a trap. Not only was Eddie there when his brother was killed, but was close enough that Eddie suffered a broken arm. The killer was never caught. All his vices that resulted from it isn't just grief, but a combination of survivors guilt and PTSD. And it's at this point that Eddie starts to change.
Though he doesn't loose his gruff demeanor, he is able to channel it in a more noble direction. But by seeing Rodger's response to his story, (it starts with Rodger, for probably the first time in the film, speaking with a subdued tone of voice, clearly grasping that there is no humor to be found from this point. By the time it ends, Rodger is in tears and admits that he'd probably act the same way if he was in Eddie's shoes.) it's here where Eddie realizes that while he KNOWS Rodger didn't commit the crime he's accused of which is why he's helping Rodger clear his name, but he's acting like Rodger committed another crime that Eddie also knows Rodger didn't do... because he's holding all toons responsible for an offense almost all of them are horrified to hear even occurred... and one all the authorities are likely going to pin on Rodger due to their similar nature.
Eddie is set up from the beginning of the story to be a character nobody would want to love... and by the end, he's the guy everyone roots for because you want to see him get the win... and he doesn't lose any edge to him. He keeps his sharp wit, gruff and jaded personality, and biting quips. That's who he is... and who he always was. And they aren't flaws. His biggest flaw is indifference and had he gotten over it, but an innocent man would have been put to death by a corrupt justice system... and the man who caused all of Eddie's pain would have done it likely with Eddie's support.
