Since she was a homunculus, Mordred's growth speed was extremely fast and she became a knight serving under King Arthur just a few years after being born. Also, because her abilities were an imitation to King Arthur's, she immediately distinguished herself as a knight.
In contrary to Morgan's intentions, Mordred admired and wished to be recognized by her father, but all that changed with King Arthur's rejection. In order to trample her father's exploits, she began to hone her poisonous fangs. Britain - which had already reached its limit all along - collapsed along with the clash of these two.
Born as a homunculus, her lifespan was short as if all affairs passed by her like a gale. Maybe for that reason, she wished to be recognized by the father who she offered all her life to. However, even Mordred was not able to guess Arthur's distress at the time.
It is simple how to deal with Mordred. Do not bad-mouth King Arthur. Do not praise King Arthur. Do not treat her like a woman. Also, do not bluntly treat her like a man. Do not behave in a stiff manner. Do not be infatuated with other Servants. Properly hear her opinions. Simple, right?