I just thought (or maybe remember) a neat proof of this fact. It involve ideas I worked on a few years ago but never published - but that's short enough so that I can explain the key ideas on MO. Let me know if you need it : I might write up the details of this proof in a short paper if you want to have a reference for it...
Fix $T$ a topos (It will be globular set) and $M$ a cartesian monad on $T$ (it will be the free $\infty$-category monad - so that the category of $M$-algebra is the category of strict $\infty$-categories).
Let $\Omega$ be the sub-object classifier of $T$. Then I claim:
Theorem 1 : $\Omega$ has a $M$-algebra structure. Moreover as a $M$-algebra it classifies the "cartesian sub-objects" of $M$-algebras.
By a cartesian subobject we mean a subalgebra $X \subset A$ such that the inclusion morphism $A \to M$ is a cartesian morphism of $M$-algebra. That is, the square witnessing it is a morphism of algebras is a pullback square.
Proof : consider $E$ the category whose objects are monomorphisms in $T$ and whose morphisms are cartesian squares. $M$ acts on this category by sending $A \subset B$ to $M(A) \subset M(B)$, and because its unit and multiplication are cartesian transformation, they give a monad structure to $M$ acting on $E$.
Now the terminal object of $E$ is $1 \to \Omega$, it automatically inherits a $M$-algebra structure and is terminal amongst $M$-algebras in $E$.
One then easily check that a $M$-algebra in $E$ is exactly a $M$-algebra with a cartesian sub-algebra and hence the theorem follows.
Remark : I had initially noticed this in the context of $\infty$-category a few years ago using the object classified instead of the subobject classifier. I used that to give a general proof that in an $\infty$-categorical context, if one defines "computads" for such a cartesian (or rather PRA) monad on a presheaf $\infty$-category then the category of computads is always a presheaf $\infty$-category. I gave a couple of talks about it in 2019 but never published the results because it seemed unclear what it was good for at the time. But it can be used to give a simple proof of the fact you mentioned.
Theorem 2 : If one defines cofibration in the category of $M$-algebra to be the left class of the weak factorization system cofibrantly generated by the $M(A) \to M(B)$ for $A \subset B$ an inclusion in $T$. Then all cofibration are cartesian monomorphism in the category of $M$-algebra.
In particular, the cofibrations of the Folk model structure are cartesian monomorphisms.
Proof : Using that $\Omega$ classified cartesian monomorphism it is easy to check that cartesian monomorphism are stable under pushout and transfinite composition. For example if $U \to V$ is a cartesian mono, classified by $\chi : V \to \Omega$, and $U \to X$ is any map, then one can define a map $X \coprod_U V \to \Omega$ that factor through to $1 \to \Omega$ on $X$ and is $\chi$ on $V$. One can then check* that this map classifies $X$, and hence that $X \to X \coprod_A B$ is a cartesian mono.
Finally, the $M(A) \to M(B)$ are cartesian monomorphism, so this conclude the proof.
(*) : ok here I'm hiding a part of the proof. To do this, One need to use that if $T_i$ is a diagram of $\infty$-category with a map $colim T_i \to \Omega$ then the subobject it classifies is the colimit of the subobject classified by the $T_i \to \Omega$. This is showed by generalizing theorem 1 to show that for every $\infty$-category $X$, the partial classifier $\tilde{X}$ is equiped with a $M$-algebra structure such that $M$-maps $B \to \tilde{X}$ correspodns to cartesian subobjects $A \subset B$ together with a $M$-map $A \to X$. This is proved exactly as Theorem 1 using the category $E$ whose object are inclusion $A \subset B$ together with a map $A \to X$ ($X$ being fixed), whose terminal object is $X \subset \tilde{X}$.
Once this is done, one can use it to show that the subobject classified by $colim T_i \to \Omega$ as the correct universal property to be the colimit of the object classified by the $T_i \to \Omega$ and really conclude the proof.
