I saw nasty bruises on the New York Times video, is there a way to avoid that generally?

The NYT video shows people fighting in tournaments, the competitiveness and associated stress levels make it easy to get bruised or injured. While protective gear is used that is by no means a guarantee that you won't get bruised or injured. It is up to you to decide if that is something you care about and up to degree.

The easiest way to avoid it is to not participate in sparring and tournaments.

Sparring is of very little importance for real world self-defense, in fact for that purpose it is counter productive. So depending on your goals and reasons for practicing HEMA you can decide on what you want to do.