Nah. We did a post on this a while back called “Your Characters Don’t Always Need To Be Good People”, a lot of our followers assumed we were talking about antagonists but it applies to both.
This gets wrapped up in the concept of “likeable characters”. A protagonist doesn’t need to be “likeable”, but they do need to be compelling. As a reader, they and their journey need to inspire interest so that we keep reading. There are plenty of characters out there (more male than female, unfortunately) who aren’t really very likeable. I wouldn’t call either Jamie Lannister, Cersei, or even Tyrion conventionally likeable (which is a personal bias) but I would say they and their circumstances are interesting, their narrative compelling. Tony Stark, especially the Stark from the Iron Man movies is another character who isn’t particularly likeable by the conventional definition. (Assume conventional likeability is on the scale of Disney Princess, Thor, and the general perception of Captain America down to Tywin Lannister, Loki, and Maleficent for negative traits.)
I’d be careful using the world “always”.
A good rule of thumb is: When working to create a protagonist with negative traits, think about a character you love with the traits you want to include in your character. Maybe this character is a villain, that’s fine if it is. Most antagonists can be redrafted into the protagonists of a different story. Maybe this character is male and you want to write a female character, again that’s fine. Men and women aren’t really that different, media just likes to pretend they are. You can easily take character traits from male characters and apply them to female characters. Most of the examples you’ll find for a traditional Heroic Journey are going to be male, but a female character can take those steps just as easily.
So, you have a character in mind. Think about why you like that character. What is it about their narrative that you find compelling and interesting? Is it their situation? Their background? The actions they take? Is it the actor’s chemistry? (Tom Hiddleston has very nice cheekbones.) What’s working for you?
Chart it all down and be honest.
Once you know why you felt that way, you can start thinking about how to replicate it. If you found these traits and that character compelling, chances are others did too and might be looking to read stories about it.
There are certain types of stories that just don’t function well with a lead character who has an overwhelming number of positive traits and few negative ones. Crime novels, particularly Private Investigators and Hardboiled Detective novels live on characters who are complicated, ethically questionable, morally repugnant, sleazy, and more than a little screwed up. Most of the conventional “good woman” tropes don’t actually work well with women warriors, especially ones who live in the gray area of grayer worlds.
Life is hard sometimes, people aren’t perfect. It’s a rainbow spectrum, there’s room for everyone. You just have to work hard to make sure that they’re interesting and worth reading about, which means understanding why you found them compelling in the first place.