Why does the Brickset inventory page incorrectly list a set as including 200 of a part when it actually includes more? One example of this issue would be a set like 42056 that has 516 black Technic pins with friction. Brickset lists it as only having 200. Is this a technical limitation of some kind? What causes this?
This doesn't have anything to do with database limitations or anything like that. Brickset scrapes their inventories from the replacement parts list on LEGO.com:

https://www.lego.com/en-us/service/replacementparts

Normally, you are able to request as many replacement parts as there are parts in the set, so this is the number that Brickset uses for it's inventory count. Unfortunately, the max you can request is 200, so Brickset doesn't list more than 200 of the same part being found in a set.