The Zemin Clan may not be weak but they certainly could not fight against the full force of the Rebel Army.
Of course, there shouldn't be any problem if the Royal Family would send their Grand Magus, Grand Knights, or Celestial Beings. The only problem is that the young King wouldn't allow such a thing to happen.
Based on the Count's words, the young King seems to be too afraid of death and he wouldn't allow his main forces to leave the capital.
This is also the main reason why the Rebel Army could slowly collect their wealth and gain more supporters.
Nevertheless, the Royal Family still sent a battalion of elite soldiers to at least show their support. Aside from that, they have lessened the taxes being paid by the clan and allowed their salt trade to expand further south of the kingdom.
It means that they could generate more money and hire their own mercenaries if needed.