I think I have found a somewhat elementary proof for the claim, (elementary in the sense that it does not invoke any big algebraic machinery), and I would like to share it in case it is of interest for anyone, and also for some feedback.
The idea is to show the injectivity of $\pi_1(U)\to\pi_1(M)$ directly, by proving that every loop in $U$ that bounds a disc in $M$, bounds a disc contained in $U$.
Let $\gamma\colon S^1 \to U$ be a loop that is contractible in $M$, therefore there exists a map $u\colon D \to M$ such that $u\vert^{}_{\partial D}\equiv \gamma$, (here $D$ denotes the unit disk).
Without loss of generality we may assume that $\gamma$ is in the interior of $U$ (it can be homotoped inward away from the boundary), that $\gamma$ and $u$ are smooth (by Whitney's smooth approximation theorem) and that $u\pitchfork \partial U$ (by Thom's transversality theorem).
Consider the preimage $C=u^{-1}\left(\partial U\right)$, this is a compact one dimensional submanifold of $D$, hence it is a disjoint union of embedded closed curves, $C=\bigsqcup_j C_j$. We denote by $D_j$ the closed topological disc whose boundary is $C_j$.
Some of the curves $C_j$ may encompass others. We call a curve $C_j$ a maximal curve if it is not encompassed by any other component of $C$. More formally, $C_j$ is maximal if there exist no other component $C_k$ and a topological disc $D_k\subset D$, such that $C_j \subset D_k$ and $\partial D_k = C_k$.
Restricting $u$ to each of the maximal curves $C_j$ yields a loop $u\vert^{}_{C_j}$ contained in $\partial U$ and contractible in $M$ by $u\vert^{}_{D_j}$. Since $\pi_1(\partial U) \to \pi_1(M)$ is injective, the loop $u\vert^{}_{C_j}$ is contractible inside $\partial U$, and we thus can redefine $u$ on each of the disks that the maximal curves bound, such that $u$ now gives a contraction of $\gamma$ in $U$.
Does this make sense?