I am by no means an expert on the Civil War, but I can give you a little bit of context from the War of 1812 and earlier and extrapolate on what I do know.
Bayonets are tricky instruments of war. There's a paradox in their use, which is that while they are much more decisive than a protracted shooting match, they are also much more risky.
The idea behind a bayonet charge is that one side or another will break and run. No one expected a mass battle or a lot of fencing. One side would advance until the other retreated or would break up because the lines they were charging didn't break.
This happened quite often, and in fact set the tactical doctrine for the British army during the War of Independence: light infantry and grenadiers in loose order, advancing rapidly across the battlefield, bayonets fixed was more often than not enough to get the undisciplined American forces to rout.
Discipline is the key factor, and we'll get back to why that's especially important in the Civil War. It took training, confidence in themselves and their officers, experience, and high morale to commit to a bayonet charge, or to commit to staying in place.
One of my favorite quotes that relates to this is from Winfield Scott, in his 1864 memoir: "The clash of bayonets, at each extremity, instantly followed, when the wings of the enemy being outflanked, and to some extent doubled upon, were mouldered away like a rope of sand. It is not in human nature that a conflict like this should last many seconds." The context is the Battle of Chippawa, which ended with an American bayonet charge.
In the same passage, Scott talks about the tactical necessities that lead to a charge, and stresses that a charge should never meet an opposing line head on, but from an oblique angle, for instance, and as you saw above, highlights the envelopment of the British flank as one of the critical aspects of his charge. He goes on at some length, but it is not necessary to quote it all.
What we can discern from what Scott is saying is that bayonet charges are dangerous. You have to be confident that your men are up to the task, because if you fail, your men breaking and running was the least harmful alternative. Much more dangerous was clumping: men who stop and, believing that it was safer to fire at the enemy huddled together, would do just that, and make easy targets for the enemy's organized volleys.
Looking at the British at Bunker Hill reveals that danger: Well-organized fire from the American entrenchments stopped the British advance cold, and small groups acted on their own, ignoring their officers, firing ineffectively at the American lines.
When the Americans at Bunker Hill finally did break, it was because of a successful bayonet charge.
So the question for officers on the field becomes: are my men capable of sustaining a charge, despite possible heavy casualties? If the answer is anything other than an enthusiastic "yes," then it would be unwise to commit to a charge.
The odds could be improved with effective artillery support, local superiority, surprise, timing, use of terrain, or any other kind of tactical advantage. The lack of any one of those, though, might doom the advance.
All of the above relates the difficulty of committing to bayonet charge when smoothbore muskets dominated the battlefield, with an accurate range of roughly 80 yards. Now imagine there are rifles on the battlefield accurate to 2-300 yards. Certainly, not every soldier is going to be a deadeye, but we're talking about the potential for massed volleys fired at massed bodies of troops, likely moving slowly forward.
The effect could be murderous even on open ground. Now imagine a small fence, a little runnel, various spots where men who lacked the force to commit might duck behind, or ten, twelve, twenty men all taking the wounded back as the rapidly diminishing force continues advancing...
You get the picture.
As for tactical reticence to use the bayonet in the Civil War, I believe, based on my limited experience in the era, that it wasn't necessarily a question of reticence, but a question of tactical impossibility. Even then, there are examples of "close with the bayonet" leading to incredible casualties. The Battle of Fredericksburg, for instance, saw Burnside throwing more and more men in the maw of a prepared defensive position despite deplorable casualties because he firmly believed that once Union forces closed with the Confederates, the Confederates would break and run.
We see the same in countless battles in the Civil War, and a few commanders even counted on it. James Longstreet, in particular, advocated operating in a tactical defense within a strategic offense: Invade the North, find a convenient defensive position, and force them to come at us.
Bayonet charges seem simple. Point your men at their men and let them loose. The reality is that they were very complex, and were difficult to pull off even under ideal circumstance with highly trained, competently led, confident troops.
I can't speak with much authority about the ratios of bayonet charges vs an affinity for a shooting match, but I would say that my casual experience suggests that bayonet charges were more common than you seem to believe, but that they were routinely defeated by tactically sound, conservative defenses. Owing to the firearms and artillery possessed by both sides, the side on defense had an enormous advantage, far more so than during earlier conflicts, and that led to difficulty in gathering up all of the advantages that would make a bayonet charge work.
Again, I'm not an expert on the Civil War, so I can't speak for sources that involve that particular conflict, but if you're interested in the use of the bayonet in earlier wars, I'd highly suggest With Zeal and With Bayonets Only by Matthew Spring. It covers the British army's tactical evolution during the War of Independence.
You can also read Winfield Scott's autobiography if you like. It's available online for free. It's kind of an entertaining work of self-aggrandizement and pomp, but it does have some interesting nuggets in there.