I understand that a large part of Napoleonic era warfare was based on the exchange of line infantry fire, however, I have seen/read about more than a few examples of line infantry marching toward the enemy without firing a single volley to engage with bayonets fixed. While the ability for infantry to fire upon the enemy is essential in this period, why would some companies not simply carry a spear or pike which would be far cheaper to make and maintain and would accomplish the same task of a bayonet?
You are definitely not alone in your thinking. The pike hadn't completely disappeared by this point, the spontoon was still commonly used by officers and boarding pikes were still used in naval combat, but there were still those dreaming of a return massed pike charges as late as the US Civil War (at the risk of linking an online source, you might be interested in this page on Governor Joe Brown's Pikes). The reasoning was basically what you expect, pikes are cheap, and soldiers armed only with bayonets couldn't possibly hope to resist a charge by weapons which are lighter and have much more reach.
Surprisingly, pikes got a lot of attention early on in the American revolution. With a lack of other arms the Americans sometimes started relying on "trench spears" to defend entrenchments and push attackers off of walls. One 1775 resolution from the Pennsylvania Council of Safety actually recommended each battalion have one or two ranks in the rear armed with 14 foot pikes, with swords and pistols as sidearms. Washington himself was generally a strong proponent of polearms, seeing them as an effective countermeasure against British cavalry. For this purpose in 1777 he ordered a large number of collapsible spears each with folding hafts to be manufactured and sent to equip Morgan's Riflemen. He also strongly enforced the carrying of halberds and spontoons by regular officers, even going so far as to forbid them from using firearms in combat since he saw it as a distraction.
Eventually though after the France and other allies began supplying large numbers of arms to the colonial army the interest in pikes died down and it isn't clear how much the weapons were actually used or what happened to them. As for why the use of pikes didn't persist, it may be in part because they were seen as old-fashioned. While Washington was working to kick-start an American halberd industry, France and England were both busy limiting regulations which prevented sergents from carrying a fusil and bayonet in the face of officers who absolutely hated the idea of going into a modern battle armed with only a sharpened stick. Probably more importantly though, despite the importance of the bayonet it was still largely a psychological weapon. Thoughout the 18th century, the amount of battlefield injuries cause by bayonets has been estimated as low as 1-2% with the vast majority being caused by bullets. Large caliber musket balls could cause some extremely terrible wounds compared to a simple blade, and the closer you are the worse they become. Coming within pike range of a loaded musket held by a man who hadn't lost his nerve was generally suicide for both you and the two or three men standing behind you. Even if they weren't defended by bayonets, charging into disciplined musket volleys was extremely difficult.
Arms and Armor in Colonial America 1526-1783
The Military Experience in the Age of Reason
The Art of Warfare in the Age of Napoleon