The trend towards two-handed weapons on western battlefields began well before firearms started to become a major force, although they may have helped hasten the shield's decline (while metal shields and armor were often proofed against pistols in the 16th century, Sir Roger Williams noted that a shield thick enough to protect against more powerful weapons like the musket would be too heavy to fight with for long periods of time). It's also true that a well armored individual doesn't necessarily need a shield to protect himself from arrows and crossbow bolts. According to one French soldier, all he needed to do was tilt his head forward a bit so that his helmet protected his face and he could advance fearlessly into a hail of arrows while they bounced off his armor. Even if a soldier was only wearing a helmet and a breastplate it became relatively unlikely for an arrow wound on his limbs to be fatal.

As for melee combat, by the late 16th century it seemed to be agreed that longer polearms held the advantage over the sword and shield in most situations. Englishman George Silver, who wrote about fencing in 1599 claimed that the two-handed sword, battle axe, half-pike, and pike all held the advantage in single combat over the sword and buckler or sword and target (a type of shield strapped to the arm), even going so far as to suggest that a half-pike might be able to fight off two swordsmen at once. In massed combat however Silver seemed to think that the sword and target could still an effective weapon. Military writers Robert Barret and William Garrard claimed that the sword and target were the best weapons for fighting in tight spaces like mines, trenches, or a breach. They also thought that they would be pretty strong weapons in a press of men, citing the effectiveness of the Ancient Roman legions. Roger Williams agreed about the mines and trenches, but wrote "The Pike is the chieftest weapon to defend, and to enter a breach" as long as it had small numbers of shot and halberders or targeters to back it up. Barret and Garrard both still only recommended around 2 or 3 targets per 100 infantry, and all the writers generally agreed that on a flat plain, nothing could resist a push of pike.

while u/hborrgg has some excellent points about the 16th century (really well done), shields started to be phased out in the 14th, which directly coincides with the time that plate armour became common on the battlefield. When one has a metal shield directly attached to ones chest, arms and thighs, they don't need a wooden handheld shield as much.

With the addition of the fact that one handed weapons are fairly ineffective against full plate, it made sense to transition to weapons that can utilize more force to disorient armoured enemies. Pole weapons provide much more force (and with spikes can sometimes pierce thin parts of armour), longswords can be used like a crowbar to pry open armour, and pikes are basically the perfect counter to cavalry; the shield, not being as useful was becoming a less common choice at this time, though still in use.