In the very first paragraph, we are directly told that by 1965 Chavez's United Farm Workers Union had gained international recognition and that this was also the year that marked the birth of contemporary Chicano theater, (lines 5-10). Thus, at the time that Chavez's United Farm Workers Union had ALREADY gained international recognition (this again is marked by the phrase "By 1965"), the Teatro Campesino has only just come to fruition.
Therefore, the Teatro Campesino could not have played a major role in the earliest efforts of the United Farm Workers Union to achieve international recognition, since during the Teatro's very first year, the Union had already achieved this.
Hope that clears things up! Please let us know if you have any other questions.