A somewhat general statement along these lines would be as follows. Suppose $\mathcal{D}$ is a cartesian closed locally presentable category and $d : \Delta \to \mathcal{D}$ is a cosimplicial object with associated geometric realization adjunction $|{\cdot}| : \mathrm{sSet} \to \mathcal{D}$, satisfying the following properties.

1. The maps $|\Lambda^n_k| \to |\Delta^n|$ have retracts for each $n \ge 1$ and $0 \le n \le k$.

    Then $\mathrm{Sing} X$ is a fibrant simplicial set for any $X \in \mathcal{D}$, so every object will be fibrant in the transferred model structure.

2. The functor $|{\cdot}|$ preserves finite products.

    Then $X^{|\Delta^1|}$ is a path object for $X = X^{|\Delta^0|}$, for any $X \in \mathcal{D}$.

This is enough to ensure that the transferred model structure exists, as described on the [nLab](https://ncatlab.org/nlab/show/transferred+model+structure#PathObjects).
In fact, this is essentially how Quillen originally constructed the classical model structure on Top--except that the category of all topological spaces does not quite satisfy most of these conditions; but it comes close enough for similar arguments to go through.

Of course condition 1 (and to a lesser extent, condition 2) is far from formal, and the kinds of model structures produced in this way are rather special.
As a trivial example, the Yoneda embedding $\Delta \to \mathrm{sSet}$ doesn't satisfy condition 1 (otherwise it would be automatic!) but obviously the model category structure on $\mathrm{sSet}$ can still be transferred across the identity functor.