Horses ridden English probably tend to be blanketed more often than horses ridden Western, partially because horses ridden in English disciplines are more likely to be boarded where there is access to an indoor arena in winter, so they can continue to be ridden and schooled regularly. This is particularly true of dressage horses. It is easier to keep a horse fit during the winter than to get him fit again after a winter off. How much winter coat a horse gets can also depend on when the horse is blanketed. If he is blanketed early (for example in mid-Sept), he will not get as heavy a winter coat as a horse blanketed in late Nov. or early Dec. But the heavier the coat, the more the horse will get sweated up if schooled hard in the winter, so a lot of horses who compete in English disciplines will be clipped in winter to remove the heavy coat so it does not take an hour of walking the horse out when the schooling is done to dry all that winter coat. But once clipped, the horse has to be blanketed to stay warm. There are a lot of reasons to blanket or not blanket a horse, just as there are multiple weights of blankets from T/O sheets (no fiberfill) to light weight T/O’s (a meager amount of fiberfill) to medium weight (a medium amount of fiberfill, & probably the most useful weight) to heavy weight (designed for horses in very cold climates, especially those in cold climates that live out 24/7.
Whether a horse needs to be blanketed or not depends on a multitude of factors that need to be taken into consideration. And yes, there is a possibility of over blanketing a horse so he gets too warm and his body starts sweating to cool him down, if the weather suddenly and unexpectedly warms up.