I'm not sure that you can separate what he says he believes from the personality he projects on air. His on air image is established not only by an emotional delivery but also by his apparent beliefs.
One would think that his ex's lawyer is pointing out the loony content of his ideas as being dangerous to the well being of the children more than his "passionate" delivery.
I am hopeful you haven't been to, and never will be subject to American Family Law courts, but I hope (and suspect) you're wrong.
It's one thing to argue, "this man scares his kids by his passionate delivery, and another to argue "his ideas are dangerous to the well being of the children".
Both are (often bogus) tactics the courts have dealt with. The former can be dealt with by requiring Jones to use a studio away from the home. The latter is a violation of free speech and likely unconstitutional prior restraint.
www2.law.ucla edu/volokh/custody.pdf
PARENT-CHILD SPEECH AND CHILD
CUSTODY SPEECH RESTRICTIONS
EUGENE VOLOKH*
The "best interests of the child" test-the normal rule applied in custody disputes
between two parents-leaves family court judges ample room to consider a parent's
ideology. Parents have had their rights limited or denied partly based on their
advocacy of atheism, racism, homosexuality, adultery, nonmarital sex,
Communism, Nazism, pacifism and disrespect for the flag, fundamentalism,
polygamy, and religions that make it hard for children to "fit in the western way of
life in this society."
Courts have also penalized or enjoined speech that expressly or implicitly criticizes
the other parent, even when the speech has a broader ideological dimension. One
parent, for instance, was ordered to "make sure that there is nothing in the religious
upbringing or teaching that the minor child is exposed to that can be considered
homophobic," because the other parent was homosexual. Another mother was
stripped of custody partly because she accurately told her 12-year-old daughter that
her ex-husband, who had raised the daughter from birth, wasn't in fact the girl's
biological father.
Courts have also restricted a parent's religious speech when such speech was seen as
inconsistent with the religious education that the custodial parent was providing.
The cases generally rest on the theory (sometimes pure speculation, sometimes
based on some evidence in the record) that the children will be made confused and
unhappy by the contradictory teachings, and will be less likely to take their parents'
authority seriously.
This article argues these restrictions are generally unconstitutional, except when
they're narrowly focused on preventing one parent from undermining the child's
relationship with the other. But in the process the article makes several observations
that may be helpful whether or not readers endorse this proposal: (1) The
best interests test lets courts engage in a wide range of viewpoint-based speech
restrictions. (2) The First Amendment is implicated not only when courts issue
orders restricting parents' speech, but also when courts make custody or visitation
decisions based on such speech. (3) Even when the cases involve religious speech,
the Free Speech Clause is probably a stronger barrier to the judge's penalizing the
speech than are the Religion Clauses. (4) If parents in intact families have First
Amendment rights to speak to their children, without the government's restricting
the speech under a "best interests" standard, then parents in broken families gener-
ally deserve the same rights. (5) Parents in intact families should indeed be free to
speak to their children-but not primarily because of their self-expression rights, or
their children's interests in hearing the parents' views. Rather, the main reason to
protect parental speech rights is that today's child listeners will grow up into the
next generation's adult speakers. (6) Attempts to allow restrictions only when the
speech imminently threatens likely psychological harm (or even causes actual psychological
harm) to children may seem appealing, but will likely prove unhelpful.
Note: I am not your lawyer, I am not even a lawyer, I am happy to give you legal advice though.