The reason why a reserve is when a formation is uncommitted is basically related to distance and time. Taking as an example Gettysburg, which is a relatively compact battlefield, the fish-hook is perhaps three miles from north to south.
If the reserve is at (say) the modern visitor's centre and is uncommitted, this means that it's ideally formed up in column and ready to react. Once a message is sent to put them somewhere, the force can move out in relatively short order to any part of the line - given marching speeds over short distances, they can start arriving in perhaps half an hour or a bit more and begin deploying.
If however the reserve is on the left flank and is deployed as a second line, then if they're needed on the right flank or at Cemetery Hill they must first "ploy" from line into column of march before they start moving. This means organizing a large number of men, and if the reserve is a corps this can involve upwards of an hour's delay. Then they can start marching, and they have further to travel than from the "true reserve" position; this can multiply the response time by a factor of three or four.
Another example of where a reserve was handled in the "classical" way for a true reserve is the Imperial Guard, especially at Waterloo. As Napoleon committed the Guard, each battalion sent into action was drawn from what were literally men standing in blocks by Napoleon's headquarters. That's why they were called "The Immortals" - they were shielded from the day-to-day casualties of combat by the Emperor, who wanted his reserves fresh for the decisive moment. (They called themselves "the grognards", or grumblers, for the same reason - they weren't getting to fight.)
One can sort of think of reserves as the "quick reaction" force; if a crisis or an opportunity develops in an arbitrary point on the field, the reserves are the force that can be sent there the quickest. It's sometimes said that battles tend to be lost by the first man to commit his last reserve, and while this is too simplistic it does have a germ of truth - that being that once you no longer have a "quick reaction" force then when the next crisis develops you can't react to it as quickly as an enemy who does have a "quick reaction" force left; conversely if a crisis develops and you haven't commited all your reserves yet but your enemy has, you can seal off the problem.
If I recall correctly, general doctrine (throughout history) was that if you had all your troops committed and no reserve, then unless your line was under heavy stress and in danger of giving way everywhere then you should draw troops back out of the line to re-form a reserve. This is interesting because at Gettysburg on July 3 there is no true reserve left - the last corps to arrive has been parcelled out as penny packets in the "support" role, which makes the individual points of the line stronger but also means that if the line is breached there's nothing to use to react to it.
(Gettysburg is an interesting battle in terms of what it means for the management of reserves, and it shows Lee using Napoleonic operational concepts related to the idea of drawing out enemy reserves and only then committing.)