I hate to beat a dead horse here, but I, too, have re-read the State Fishing Regulations. According to it and the definitions of game fish, rough fish, and minnows, the Grass Pickerel is a game fish. You were right about that. Please see the image below from page 13 of the regulations. If you read further, however, you will find that there is
no open season for pickerel.
That means that it would be illegal to catch them. Please see below the excerpt from page 50, the start of pages describing the Open Seasons, Length Limits, and Bag Limits for game fish. It clearly states that "Fish Species not listed have
no open season". Also see below that a copy of pages 50 and 51 which list the state-described game fish and their Open Seasons, Length Limits, and Bag Limits.
Page 13 - Game Fish, Rough Fish, and Minnows; Definitions

Page 50, Excerpt - Open Seasons, Length Limits, and Bag Limits

Pages 50 & 51 - Open Seasons, Length Limits, and Bag Limits