One principle alone that that would help him here:

Principle 2: Not bad teams, bad leaders

[Leaders] must face the facts through a realistic, honest assessment of themselves and their team’s performance. Identifying weaknesses, good leaders seek to strengthen them and come up with a plan to overcome challenges… It starts with the individual and spreads to each of the team members until this becomes the new standard."

What the leader does has a ripple effect on the behavior of everyone else in the team. When Passive/Defensive leaders bury their head in the sand about poor performance, those poor behaviors flow on to the rest of the team. Aggressive/Defensive leaders punish and blame which leads to inaction or internal conflict. Constructive leaders guide and direct and expect team members to take responsibility to meet the expected standards.

As you said, it really is simple for Zach to own this in public, and address everything in private, during team and position meetings. Not throw your teammates under the bus in the press conference.

Principle 4 isn't that far behind though: Check the Ego

Ego clouds and disrupts everything: The planning process, the ability to take good advice, and the ability to accept constructive criticism. Often the most difficult ego to deal with is your own… When personal agendas become more important than the team, and the overarching mission’s success, performance suffers and failure ensues."

There is nothing wrong with winning, or wanting to win. The problem is when someone’s drive to win and standout comes at the expense of teammates and the organization. Competitive leaders aim to win, Achievement Leaders aim for performance excellence. To drive competitive instincts, Competitive leaders frequently pit staff [team mates] against each other – this ultimately leads to dysfunction and performance suffers.