The short answer is that soldiers did continue to fight with swords and other close-combat weapons for hundreds of years into the Gunpowder Age.

Until the late 1600s-early 1700s, most European infantry armies were composed of a mix of arquebusiers/musketeers, pikemen, and various other light infantry. The Spanish "tercio" square unit is a famous example, being divided into thirds of gunners, pikemen, and swordsmen. The gunners caused damage at range, the pikemen protected the gunners from cavalry attacks, and the swordsmen could slip under and around enemy pikes during a skirmish or be deployed flexibly as needed. The tercios dominated the battlefield in the late 1500s, but gradually were phased out as field artillery improved and units were deployed in thinner rows and columns less susceptible to artillery.

In the German states of the late 1400s-1500s, a typical mercenary unit was composed of Landsknecht, soldiers armed with pikes for dealing with cavalry (and other pike formations), large swords for infiltrating pike squares, and arquebuses for range.

As firearms technology advanced, it became cheaper for the increasingly-centralized monarchical armies of Europe to arm their poorly trained, less professional soldiers with guns capable of killing heavily armored professional soldiers. The innovations in mobile field artillery brought about by Swedish King Gustavus Adolphus II during the Thirty Years War (1618-1648) also helped tilt the balance toward fast-moving musketeer units and away from slow-moving massed pikes (easy targets for small cannon), requiring fewer swordsmen to deal with the pikes.

It's difficult to generalize several hundred years of European military development, but the final component leading to the transformation of European infantry armies into the mostly-musketeer forces of the later 1700s was the invention of the socket bayonet at the end of 1600s. Earlier bayonets were plugged into the barrel of the gun, making them useless as anything but a spear at the last moment before melee combat. The socket bayonet allowed a pikeman and a gunner to be merged into a single soldier.

Due to the slow rate of fire mentioned in the question, bayonet charges by musketeers or riflemen continued to be an effective and frequently-employed tactic until the modern era of warfare.

Sources:

John Keegan, A History of Warfare (1993)

Russell F. Weigley, The Age of Battles: The Quest for Decisive Warfare from Breitenfeld to Waterloo (1991)

I've fielded similar questions here, here, and here.

For a short and more specific answer to your question, the reason that muskets replaced other weapons is because they were effective, and they complimented the developing military complexity and standards for order and discipline extremely well.

To break it down a little more, muskets were effective weapons, period. They were not nearly as unreliable, slow, or inaccurate as is often represented in pop culture. A trained infantryman would be able to fire as often as once every 15 seconds (give or take), and the weapon itself was consistently able to strike a stationary man-sized target somewhere on the body at 80 yards or so. Multiply that by a formation of hundreds, and most targets, even armored cavalrymen, could be torn apart. The cleaning and maintenance of a musket was also not more or less difficult than any other ranged weapon, and they could be mass produced on an industrial scale, as opposed to a cottage industry that produced bows, for instance.

Muskets are a weapon that compliments and supports close-formation fighting and don't make for great individual weapons. Even if 15 seconds is faster than we expect, your average soldier probably isn't going to stop and reload in a 1-on-1 situation. But in a group, where individual acts of valor matter less than just keeping in formation, it works quite well. The relative simplicity of using a musket means that commanders can spend more of their (very little) training time on battlefield movement, which allows for more complex and arguably useful options in combat.

I go into a lot more detail in the above linked posts, but I'm willing to answer follow ups, too.

I believe you are projecting a single person capability on the troop tactics. Early muskets took a bit to load, partially due to unnecessarily complex routines evidenced by the contemporary field manuals. The simplest way of shooting was the cascade fire, used by e.g. Polish and Hungarian infantry in 16th and 17th century, where the last rank was shooting over the heads of their kneeling comrades. After the volley, the last rank was starting to load, shooters in the rank in front of them were standing and firing and so on. Western armies quickly adopted the countermarch, which is a similar technique, but with the foremost rank firing and then moving to the back of the unit. In any case, given that the musketeers were usually arranged in 10 ranks, when the last one fired their salvo, soldiers in the first rank have already loaded their were able to fire again, allowing for more or less continuous fire. It should be noted that if the amount of powder carried by an average shooter is any indication, each musketeer was expected to fire around a dozen times per average battle, so the frequency of fire could not have beet too high, giving plenty of time to reload.

Now, returning to original question. As others already mentioned, musketeers by all means were carrying side weapons and sometimes used them in combat, although the rising popularity of pikes made them hardly useful, leading to separation of roles between musketeers from pikemen. That said, pikemen often carried side weapons to use them at a last resort. This changed in the 18th century, when the introduction of the bayonet gave musketeers additional protection from cavalry and capability of fighting in close quarters, rendering hand weapons redundant, especially given that 19th and 20th century sword bayonets were essentially short swords in their own right.