That's not true. If that was the case then breeders would shoot for Sept, Oct, or Nov. Giving their horses an advantage in an age event. I was just talking about this very thing with a cutting horse trainer friend in TX last week. There have been times when breeders shot for a birth during the 1st week in January but end up in the last week of December thus having their foal "turning" 1 in one week or so. Many breeders shoot for foals that are born during the last week of January, or February or March.
The January 1st birthdate only counts for performance horses in age required events. I've had horses that were born in April, May, and as late as September. As long as there isn't an age requirement at a show, it doesn't matter. As for training, since I don't worry about age requirements, I go by actual age, not January 1st.