I know it melts faster, but tearing cheese into rough parts and adding it to a white sauce melts just fine. Is there a specific reason to why you should shred cheese before adding it to a sauce?
You're right that the smaller pieces of cheese will melt faster than a whole block when added to a sauce.
The main advantage to shredding or grating cheese is that you create smaller pieces of uniform size, and often smaller than can be achieved just by crumbling (at least when working with harder cheeses).
The smaller the pieces the faster they melt, and if they're a consistent size/shape you don't have to investigate the sauce closely to find when the last, largest bit of cheese has been incorporated. This helps ensure the cheese will be evenly distributed throughout the sauce.
Lot's of small bits (of anything) makes for more surface area, therefore melting not only faster but evenly.
Dissolving works the exact same way.
If you put cheese in the last step of cooking, it would melt in perfect timing. If you have to cook longer with other ingredients, for example: pasta, since pasta serve al dente, it would overcook quickly. Also, shed cheese would give even taste to the dish.
It depends on what you are putting the cheese into. If it is sauce then it would be fine.
It may take less time to locate and use a grater than to break the cheese by hand. So theoretically, you could use a knife to slice the cheese up and it wouldn't make a difference in the quality of the sauce. It just might take longer to cut it up.
Other answers mention it helps the cheese melt faster and more evenly. This is, truly, one great benefit to shredding or grating the cheese.
One other factor is moisture.
Especially with drier cheeses, just trying to melt them with heat can easily end up rubbery bits and oil, as the proteins seize up and the fats separate. Wetter cheeses melt better, and with drier cheeses this can be approximated by adding liquid - though smaller shreds and longer soaking help get the moisture into the cheese evenly. Melting them into just liquid like milk, though, doesn't always work as planned - the liquid is too thin to hold the cheese, so it clings to itself instead of spreading into the liquid and when it hits the right temperature, it seizes, clumps up with the same rubbery oily result.
Melting into a sauce works better, since it has enough stuff in the sauce for the cheese to cling to, and it spreads the cheese out into the sauce and hydrates it, which lets it melt very nicely into the sauce. The result is smooth, not tangled.
In short, the smaller shreds also work mechanically - the cheese is spread out quickly and surrounded by the sauce, instead of staying in cheese-clumps, and with better access to the moisture as it heats will tend to mix out into the sauce instead of cling to itself (into aforementioned rubbery lumps and oil).
With larger chunks, the edges and the surface are exposed to the sauce and should melt nicely. The inside, though, doesn't have nearly as much exposure - it is possible, if the temperature is a bit on the lower side and the stirring frequent enough, for the time it takes those outer edges to be melted and the mixing pulls them into the sauce, to be enough time for the next layer to be moistened and heated to melting. It is also possible, if the heat is higher or the mixing a less diligent, for the insides of the cheese to not get exposed to the sauce quickly enough, and while the outer layers will melt into the sauce correctly, the inside doesn't have enough exposure to the moisture, will stick to itself rather than the sauce, and end up rubbery chunks and oil. Again.
So shredding the cheese works on several levels. One is that it lets the cheese have very good exposure to moisture, which helps it melt instead of solidify. Another is that it makes it easier for the cheese to grab onto the solids in the sauce and spread out, rather than clump up by only grabbing other shreds of cheese. A third factor is that shredded cheese needs less stirring - since the shreds melt quickly, and a few stirs will roughly mix them into the sauce. With larger chunks, stirring has to continue longer since each layer of cheese that gets melted off the chunk needs to be stirred into the sauce, to prevent said clumping, and the inner layers will of course be slower to heat and take longer to melt than the outer (hence more stirring).
A last factor is, the texture changes as cheese is added (of course) and shreds will be easier to portion out to mix gradually. A thinner sauce can be trickier to get the cheese to melt into, and it helps to have less cheese added at once and stir very well, to maximize exposure to the sauce. As it gets thicker, from the cheese previously melted in, it is easier to add more cheese in at once and it melts in better (including somewhat bigger pieces, at this point). It is, of course, easier to separate and on-the-fly adjust the amounts of cheese for this sort of gradual addition with piles of small shreds rather than fewer bigger chunks.