Bayonet charges happened only after the unit being charged was weakened through musket and artillery fire. To attempt to charge before that happened would result in catastrophic casualties for the the chargers as they suffer the inevitable volley from the stationary unit at painfully close range.

Bayonet charges were not uncommon however it is true that protracted melee's were rarely the result. It often took a great deal of manoeuvring and care to ensure that a unit was able to enact a bayonet charge properly without being swept away by cavalry or by becoming lost due to disorder following a charge. It is, as with almost all things in the warfare of the period. A question of morale and order. A bayonet charge is terrifying for everyone involved.

To quote some soldiers of the period

the Surgeon George Guthrie

for the best possible reason, that one side turns and runs away as soon as the other comes close enough to do mischief.

John Mitchell

...in some scrambling attack of works (defences), or a hasty flight out of woods or villages, [that] a soldier may, perhaps, have been killed or wounded with a bayonet is possible, but to suppose that soldiers ever rushed into close combat, armed only with bayonets, is an absurdity, it never happened and can never happen.

So the soldiers of the era were well aware of the purely psychological nature of the bayonet charge. Considering protracted melee with bayonets a rarity if not an impossibility.

For example, in the battle of Maida there was a collision between a French unit and Kempt's light Battalion. The French fired a few ineffectual volleys at a few hundred yards. At 200 yards Kempt ordered his men to halt and remove their knapsacks in preperation for a charge. The French took this as a sign of hesitation, that the British were wavering and not ready to fight. They fired a volley and then charged. However when the British responded in kind they immediately broke and ran. They had been expecting men ready to flee and to find them ready to fight broke them.

There are a few instances of protracted bayonet fights occurring. In July 1814 a light company was driving away some French skirmishers from the top of a hill. As they arrived they came face to face with a French column of infantry just coming over the hill. They charged them and such was the disorder and panic they caused that they drove the French of the hill after a good deal of killing. (To the extent that one soldier reported to his officer with only the bloodied broken stub of his bayonet, having broken it off in his enthusiasm.)

The British tactic during the Napoleonic wars is very interesting. The Battalion would sit on the reverse slope of a hill in two ranks, protected from cannon fire. The light company would be spread out on the other side of the hill to cover them from enemy skirmishers.

The French column would then advance up the hill, screened by skirmishers. The British skirmishers would then withdraw as the French approached the top. The British line would then rapidly advance over the top of the hill and fire one or two rapid volleys into the French as they desperately tried to deploy into line to return fire. The British would then charge down the hill at the French who then frequently broke and ran, demoralised as they were by the sudden appearance of soldiers directly in front of them, sustaining a close range volley or two and then being charged. However, it was considered dangerous to halt an advance to fire another volley, as it would interrupt the momentum of the advance and damage the morale of the men. This was laid out in the official manual for the British army and an intrinsic part of strategy.

The French on the other hand would bombard the enemy with vast quantities of artillery fire, softening them up. The French columns of infantry would then advance en screened by swarms of skirmishers who would then also take a toll on the enemy lines. When sufficiently weakened the French would advance close enough to deploy into line and deliver a few volleys at close range before charging a demoralised unit that has suffered at that hands of artillery and skirmishers and is wanting to run. Such is what happened at Ligny.

The bayonet charge was frequently the culmination of an attempt to damage the morale and order of the enemy infantry. With normally one side not being able to stand, occasionally though you would get two reasonable confident units thrust upon each other that would engage in melee lasting slightly longer.

Why is a charge so much more terrifying that getting shot at though? Many men will stand getting shot at when they will not stand getting charged. It may simply be because, as William Napier put it,

the moral influence of [the bayonet] is great. Men know, psychologically and physiologically, that ... it will prick their flesh and let out life, and therefore they eschew it. Many persons who will stand fire will not stand a charge, and for this plain psychological reason - that there is great hope of escape in the first case, very little in the second, and hope is the great sustainer of courage.

British Napoleonic Infantry Tactics 1792-1815 - Philip J Haythornthwaite