This is a section that will explore some of the misconceptions I've seen about Hitsugaya over the course of the series. These are my opinions in response to other opinions I've seen around, and no one person's opinion has more value than another's.
❝Hitsugaya has a cold personality.❞
It should be noted that Hitsugaya is not a cold character. Despite his affinity for ice, his personality does not necessarily reflect the element in which he thrives. As seen on the personality page, he has several rather fiery character traits. If I had to choose one word to describe him, I would choose the word "passionate." He comes across as cold to those around him, and certainly has his moments where he actually is, but it is not the core of his character in the slightest. Let's look into this misconception a little bit.
As a child, Hitsugaya is ostracized by the children around him. He thinks it is due to his appearance, but also notes that others think he is "cold." Everyone aside from his grandmother and Hinamori seem wary of him, and are eager to avoid him or limit their contact with him. This likely has very little to do with his personality at all, though even as a child he is haughty and snarky. I believe it has to do with instinct. Hitsugaya is a powerful soul with an overwhelming reiatsu, and as a child he is unaware of this power and therefore entirely unable to control it. As we witness with Ichigo, if one's own reiatsu is strong it can affect those around them. We see time and time again that a powerful reiatsu is enough to bring someone to their knees or physically makes them ill. And the person releasing the powerful reiatsu is entirely unaffected by it because it is simply an extension of them.

Hitsugaya is perceived as cold because of his powerful, ice-based reiatsu that he initially cannot control. This elemental reiatsu is powerful enough to physically affect those it touches, lowering the temperature in his immediate vicinity. Those around him avoid him because this reiatsu is activating their survival instincts - they sense something different, something powerful, something intimidating, something potentially dangerous... something cold. It is his power they are sensing, though they do not know that and find a way to rationalize their instincts by finding flaws in his character.

There is not one person that actually knows him that has described him as or would consider him cold. His grandmother knows exactly the type of person he is and loves him with all of her heart. Hinamori is his best friend and is not intimidated by him in the slightest, even before he was able to control his reiatsu. This could be partially attributed to the fact that Hinamori herself has strong reiatsu and could therefore be less affected by his, but she too is aware of his true personality and remains friends with him even when she is enrolled in the shinigami academy and he is not.

It is shown numerous times through his interactions with his companions that he is not an overall cold person. He gets angry quite frequently, and he cares for certain people more than he cares about himself. He is fierce in battle and fierce in his devotion to those important to him and his role as Captain of the Tenth Division. These are all attributes typically possessed by those with a "warm" personality. There are times when Hitsugaya is, indeed, cold. These moments are mostly reserved for his enemies or people he distrusts, which is a normal facet of literally anyone. The only ones to call him cold throughout the series are those he directs his icy reiatsu toward - his foes. This is not indicative of an overall cold personality. Would you be open and yourself around someone you don't like or trust? Are you inclined to drop your guard in front of someone who wants to kill you? Most people would say no. Hitsugaya, in all of his battle prowess and quick wit, despite being a shinigami and magically wielding ice, is still a normal person. Normal people are not defined by one singular personality trait or flaw, and they are certainly not defined by the stereotypes of their professions, hobbies, or skills.
❝Hitsugaya is weak for a Captain.❞
I won't deny it - we see Hitsugaya get his ass kicked on multiple occasions. He is built up as this shinigami prodigy, with the most powerful ice-elemental zanpakutou, and yet...? His opponents always seem to be stronger than him, and even when he wins, he is usually terribly wounded. We don't see him comfortably wipe out a single enemy, so he must be pretty weak for a Captain, right? Wrong. You know who else we see always getting his ass kicked, fighting opponents stronger than him, and getting grievously wounded? Kurosaki Ichigo, our resident main character, the hero of our story. Kurosaki Ichigo, underdog champion.
Perhaps this misconception begins with the way Hitsugaya is introduced to readers. He is one of the first Captains introduced in the story that is not introduced as an antagonist. Instead, he is considered a protagonist from the very start, investigating his fellow villainous shinigami for what he accurately believes to be a terrible conspiracy. We see that he ends up on the side of the protagonists long before they ever actually meet him. Now, let's think about the Captains we consider the strongest and most terrifying for a moment. There is, naturally, Zaraki Kenpachi, who is horrifying as all hell and can probably kill with his reiatsu alone. There is Kuchiki Byakuya, who will cut you down without taking a single step. There is Captain-Commander Yamamoto, who can probably set you on fire with a wave of his hand even if you are twenty miles away. These are terrifying shinigami, right? They are. And they were all introduced as antagonists.

These three Captains are seen as obstacles that Ichigo must overcome if he wants to save Rukia from being executed. They are insanely powerful and he has to defeat them despite overwhelming odds. It takes many, many chapters for Ichigo to prepare to defeat Zaraki and Byakuya, and many more to actually do so. He never gets to fight Yamamoto (thankfully, because at that point there would be no way in hell that he'd win) but the Captain-Commander is still an antagonist until he learns the truth. We think they are terrifying because we saw them as opponents through Ichigo's eyes. If Hitsugaya had been one of Ichigo's opponents, I'm sure he would have had a hell of a time against him too.

Hitugaya is not weak in the slightest. He is simply presented as an underdog protagonist, just like Ichigo, who is forced to fight and defeat opponents much stronger than himself. Unlike Ichigo, he is not the main character, so we don't normally see his growth in strength until his next battle, which will be against an opponent much stronger than him, etcetera, the cycle continues. Arguably, he is even stronger than Ichigo, at the very least emotionally. While Ichigo is shown to give in to despair many times throughout the series, albeit temporarily, Hitsugaya never does. He never quits, never backs down from a challenge no matter the odds, and never lets himself succumb to defeat and melancholy. He is a protagonist in his own right, and could very easily have had his own series as the main character.