Henry VI really is a sweet guy. He's always trying to make peace among his feuding nobles—and boy, does he have a lot of feuding nobles. He seems virtuous and good-natured, and he readily listens to Gloucester's generally solid advice. He's such a do-gooder, in fact, that in response to his marriage prospects he tells Gloucester, "I shall be well content with any choice / Tends to God's glory and my country's weal" (5.1.26-27). And this seems to be what he wants most of the time: God's glory and his country's good.
The trouble is, while Henry's totally a guy you'd want for a friend, he may not be the guy you'd want for a king, at least in the rough and tumble world of Medieval politics as Shakespeare presents them. His father, Henry V, and his grandfather, Henry IV, were both men of action—real get it done types bursting with macho. They brought the brawn, military charisma, political savvy, and insistence on getting the job done whatever the cost in spades. Big boots these guys left to fill, to say the least. And to continue the metaphor, Henry has small feet.
Henry VI does show some of his father's talent for speechmaking, but he's not so strong on the decisive action. He mostly uses his rhetorical talents to try to make peace, which is an admirable goal, but not likely to impress his nobles, who are trying to win a war against France. Check him out in action:
Uncles of Gloucester and of Winchester,
The special watchmen of our English weal,
I would prevail, if prayers might prevail,
To join your hearts in love and amity.
O what a scandal is it to our crown
That two such noble peers as you should jar?
Believe me, lords, my tender years can tell
Civil dissension is a viperous worm
That gnaws the bowels of the commonwealth. (3.1.69-77)
Okay, so Gloucester and Winchester are major twerps when it comes to bickering with each other. It's like they can't even help themselves—put these two in the same room and an argument is guaranteed to ensue. So here Henry makes a good little speech asking these two uncles of his to bury the hatchet. But in the same speech, he acknowledges his lack of experience (hey there, "tender years"), which totally undermines his authority. And remember: They may be older and his uncles, but Henry is totally their boss.
As for that noise within? Yeah… that would be Gloucester and Winchester's men beginning to fight. In the streets. While Henry stands there and basically asks the two feuding men to pretty please play nice. And later in the scene, when a riot erupts, Henry needs help quieting things back down, which is not something his dad or grandfather would have needed at all. So not only is Henry slow to act, but he's not awesome at acting once he finally does.
He also makes a few pretty bad calls, partly because he's too inexperienced to know better and partly because he's just not the political type. For instance, he pins the red rose on after attempting to resolve the argument between Somerset and Richard (for more on this, check out the "Symbols" section), and he also dumps a rich and well-connected woman in favor of Margaret, who's poor and less connected. This is one of the few decisions Henry refuses to listen to Gloucester's advice on—despite having not met either woman himself.
Henry's many things, Shmoopers, but he's just not the clever and ruthless king type. And in the few instances where he is stubborn—the Margaret decision is a standout—his insistence is misguided and ill informed. In his defense, though, he is pretty new to this whole king business.
You can make a solid argument that it's morally better to be a good person than a talented king, and maybe the political world in which Henry VI finds himself is deeply flawed. But you can see why the loyal—but savvier—Gloucester is worried about Henry's ability to lead England. Henry's heart can be in the right place all day long, but until his head is, things remain dicey for England.