By Dan Burns (Reuters) -U.S. union membership rates fell to fresh record lows in 2023 despite it being a year of headline-grabbing organized labor strikes from the Rustbelt to Hollywood and some continued organizing successes at companies such as Starbucks. The union membership rate fell to 10.0% from what had already been a record-low 10.1% in 2022, the Labor Department said on Tuesday in an annual census of the U.S. organized labor landscape that also showed continued growth in union membership among people of color and a narrowing pay advantage for union workers. The number of union members…