The word "during" implies that you are referring to a period of time so it is okay to say "During the Chinese Spring festival, families get together." The word "on" doesn't necessarily refer to a period of time so we must add information to make that clear. This is a correct example: "On the day of the Chinese Spring festival, families get together." Adding "the day" after "on" makes it clear that you are referring to the period of time.

Thanks

During is better. On is specific to a day, but Chinese New Year is a period of time lasting over a week. Ex: During the holiday season, families get together/During Chinese New Year, families get together. (a period of time) On Christmas Eve, families have dinner./On the eve of Spring Festival, families get together. (a day)