I think that one can prove the relative version of Ehresmann's lemma (and also versions for manifolds with corners, which is my interest) without any technology as follows:

- Let us first recall the following simple proof of Ehresmann's lemma:

Let $U\subset B$ be a coordinate neighborhood $U\simeq \mathbb{R}^n$ centered at $b\in B$.
Consider the coordinate vector fields $\partial_1,\dotsc,\partial_n\in \mathfrak{X}(U)$, and pick their smooth lifts $X_1,\dotsc,X_n\in\mathfrak{X}(\pi^{-1}(U))$ under $\pi$, respectively.
This is possible if and only if $\pi$ is a submersion (more below).
The properness of $\pi$ implies that $X_i$ are complete, so they admit complete flows $\phi_i\colon \pi^{-1}(U)\times \mathbb{R} \to \pi^{-1}(U)$. Let $f\colon \pi^{-1}(b)\times U \to \pi^{-1}(U)$ be the map defined for all $\xi\in \pi^{-1}(b)$, $t=(t_1,\dotsc,t_n)\in U$ by
$$
f(\xi,t) \colon =(\phi_1^{t_1}\circ\dotsb\circ\phi_n^{t_n})(\xi).
$$
We have $\pi(f(\xi,t))=t$ because $X_i$ lift $\partial_i$, and $f$ is a diffeomorphism with inverse
$$
f^{-1}(x) = (t, (\phi_n^{-t_n}\circ\dotsb\circ\phi_1^{-t_1})(x)),
$$
where $t=f(x)$.

- The relative version of Ehresmann's lemma---given a submanifold $N\subset M$ and assuming that both $\pi\colon M\to B$ and its restriction $\pi|_N\colon N\to B$ are submersions---can be obtained by choosing lifts $X_1,\dotsc,X_n$ of $\partial_1,\dotsc,\partial_n$ that are
**tangent to $N$** as follows:

Denote $s:=\dim(N)$, $m:=\dim(M)$. For each $x\in \pi^{-1}(U)\cap N$, let $V_x\subset M$ be a coordinate neighborhood $V_x\simeq \mathbb{R}^m$ of $x$ such that
$$
V_x \cap N = \{ (x_1,\dotsc,x_m)\in V_x \mid x_{s+1}=\dotsb =x_m = 0\}.
$$
The existence of such "slice charts" for a submanifold $N\subset M$ is a standard fact.
Because the differential $(D\pi)_x\colon T_x N \to T_{\pi(x)}B$ is surjective, we have $n\le s$, and we can pick $n$ coordinates, w.l.o.g. $x_1,\dotsc,x_n$, such that the restriction
$$
(D\pi)_x: \mathrm{span}\{\partial^x_1,\dotsc,\partial^x_n\} \to \mathrm{span}\{\partial_1,\dotsc,\partial_n\}
$$
is an isomorphism. This is a simple fact from linear-algebra, namely, that any generating system contains a basis.
By shrinking the neighborhood $V_x$ if necessary, we can assume that the above isomorphism holds for $(D\pi)_y$ for all $y\in V_x$.
For each $y\in V_x$ we can now define the vectors
$$
X_{x,i}(y) := ((D\pi)_y|_{\mathrm{span}\{\partial_1^x,\dotsc,\partial_n^x\}})^{-1}(\partial_i) \in T_y M.
$$
We obtain smooth vector fields $X_{x,i}\in \mathfrak{X}(V_x)$ that lift $\partial_i$ and are **tangent to $N$**.
We proceed similarly for $x\in \pi^{-1}(U)\backslash N$ to obtain neighborhoods $V_x$ and vector fields $X_{x,i}\in\mathfrak{X}(V_x)$ that lift $\partial_i$.
Pick a cover $(V_\alpha)\subset \{ V_x \mid x\in \pi^{-1}(U)\}$ of $\pi^{-1}(U)$ with a subordinate partition of unity $\lambda_\alpha\colon V_\alpha\to [0,1]$.
Define $X_i := \sum_\alpha \lambda_\alpha X_{\alpha,i}$.
Then $X_i\in \mathfrak{X}(\pi^{-1}(U))$ are smooth vector fields that lift $\partial_i$, so they can be used in 1 to construct the trivializing diffeomorphism $f$.
Because $X_i$ are moreover tangent to $N$, their flows through points in $N$ remain in $N$ for all times, so the diffeomorphism $f\colon \pi^{-1}(b)\times U \to \pi^{-1}(U)$ restricts to the diffeomorphism
$$
f|_N\colon \pi^{-1}(b)\cap N\times U\to \pi^{-1}(U)\cap N.
$$
This proves the relative Ehresmann lemma as stated in the question.

- Analogously, one can prove Ehresmann's lemma in the case that $M$ is a manifold with corners.
Here one has to assume that the restriction of $\pi$ to any open codimension-$k$ boundary strata of $M$ is a submersion.
Given $x\in M$, one picks a corner chart on $M$ that intersects only those strata that contain $x$, and one chooses the coordinates $x_1,\dotsc,x_n$, which were used to define $X_{x,i}$, in the strata of highest codimension.
In this way, the vector fields $X_{x,i}$ are tangent to any strata containing $x$.