Does the traffic coordinator have the right to issue a ticket?

I don't think so. The traffic law doesn't say they have authority, but it doesn't say they don't. This may be a legal loophole. The following is the implementation of the punishment in the traffic law:

Chapter IV Execution of Penalty

Article 18 Where a party is fined, it shall inform the party to pay the fine at the designated collection agency within 15 days from the date of receiving the penalty decision. If the party fails to pay the fine within the time limit, it shall be fined at 3% of the fine amount every day.

Nineteenth penalties for motor vehicle drivers in any of the following circumstances, you can collect fines on the spot:

(1) Impose a fine of 20 yuan (inclusive) on motor vehicle drivers who cross the border outside the jurisdiction (bounded by municipalities directly under the Central Government, the same below), and the traffic police will make a decision on punishment on the spot;

(II) Impose a fine of more than 20 yuan on motor vehicle drivers who have not passed through the area in remote and inaccessible areas and expressway, and it is indeed difficult for motor vehicle drivers to pay the fine to the collection agency where the traffic violation occurred, and the motor vehicle drivers take the initiative.

Twentieth for non motor vehicle drivers, pedestrians and passengers fined twenty yuan (inclusive) or less, the traffic police made a decision on punishment, on the spot.

Twenty-first collection of fines on the spot, it shall issue a fine receipt issued by the provincial finance department to the parties.

Twenty-second fines collected by traffic police on the spot shall be handed over to their subordinate units within 24 hours from the date of collection. The unit shall, within two days from the date of receiving the fine paid by the traffic police, pay the fine to the collection agency.

Article 23 If a motor vehicle driver's license issued in a different place is temporarily withheld or revoked, the traffic administrative department of the public security organ that made the punishment decision shall transfer the motor vehicle driver's license and the punishment decision to the traffic administrative department of the public security organ where the motor vehicle driver's license is issued within 15 days after the punishment decision takes effect.

Twenty-fourth if a party refuses to accept the punishment decision made by the traffic administrative department of the public security organ and its traffic police, it may apply for administrative reconsideration or bring an administrative lawsuit according to law.

Twenty-fifth if a party refuses to pay the fine for not fulfilling the punishment decision, in addition to these provisions, it may also impose administrative detention on the party or apply to the people's court for compulsory execution according to the regulations on administrative penalties for public security.