The demonstrated cross wind component is only that supplied by the manufacturer given the actual conditions used to achieve what they thought a useful limit might be.
The manufacturer does not choose a minimum value, it is in the design standards, which were quoted on the first page of this thread. Agreed that they could certify a stronger crosswind capability if they choose, but what's in it for them to take on the added liability.
If an aircraft operator were to "set" a different value, that would be their choice, but probably arbitrary, and difficult to support with test data. But yes, I can see value in telling new solo students that a given crosswind was too strong. That would not be because of the aircraft, but rather that that new pilot had not yet achieved the standard of "average pilot skill" stated in the design requirement. Otherwise, the value in the flight manual would seem to be a great reference for the average pilot of the aircraft. I cannot see a reason why an operator would state a greater value, just be silent on that.
The design requirements to which the plane was certified overtly do not prohibit operations in stronger crosswinds. I opine that this would be because the regulator or manufacturer logically do not want to tell a pilot that they cannot land a plane legally - 'cause it's going to happen before or at the end of the fuel! As a regulator, I would much rather that the interpretation by the pilot results in them going off the side of a runway due to too strong a wind during a landing attempt, than a crash off airport because they thought that a landing attempt there would be prohibited. Obviously, a nearby airport with a different runway orientation would be preferable - if there is one. But I have certainly flown to one runway airports where the actual crosswind was stronger than forecast, and the next nearest runway was hundreds of miles away! You're landing on that runway or none at all!