I don't think that what I will now write is the best or quickest answer to your question, but maybe it is worthwhile to set it down because the point of view is a bit of useful folklore.

First let me translate the problem from Morse theory to Hamiltonian dynamics:

1. Instead of the submanifold $L$ consider its conormal bundle $\nu(L) \subset T^* M$ defined as the set of all covectors bases at some point of $L$ that vanish on the tangent space to $L$. The conormal bundle is a Lagrangian submanifold.

2. Consider the geodesic flow $\phi_t : T^*M \setminus O \rightarrow T^*M \setminus O$ on the slit cotangent bundle. Consider also the flow $$ D\phi_t : T(T^*M \setminus O) \rightarrow T(T^*M \setminus O) $$ obtained as the differential of the geodesic flow.

3. Last, but not least, consider the bundle of tangent Lagrangian planes over $T^*M \setminus O$ that I'll denote by $\lambda(T^*M \setminus O)$.

The flow $D\phi_t$ induces a flow on $\lambda(T^*M \setminus O)$ that I'll denote by the same symbol. This flow has an important "twist" condition: if $p_x$ is a point in $T^* M \setminus O$ and $\phi_t(p_x)$ is the integral curve passing through it, consider the vertical Lagrangian tangent plane $V_{\phi_t(p_x)} \subset T_{\phi_t(p_x)}(T^* M \setminus 0)$ and the curve of Lagrangian planes in the (symplectic) vector space $T_{p_x} (T^*M \setminus 0)$ defined by $$ t \mapsto D\phi_{-t}(V_{\phi_t(p_x)}) =:\Lambda(t). $$

4. The twist condition satisfied by geodesic flows for Riemannian and Finsler metrics (among other non-degenerate Hamiltonians) implies that if $t$ and $t'$ are sufficiently close, then $\Lambda(t)$ and $\Lambda(t')$ are transverse.

5. Now we have to translate focal points into this language: Consider $p_x$ on the conormal bundle of $L$ (at the point $x$) and the orbit $\phi_t(p_x)$ $(t \in \mathbb{R})$.

Proposition. The point $y$ obtained by projecting $\phi_s(p_x)$ onto $M$ is a focal point if the Lagrangian plane $\Lambda(s) \subset T_{p_x}(T^*M \setminus O)$ does not intersect the (Lagrangian) tangent plane of the conormal bundle $\nu(L)$ transversely at this point. The dimension of the intersection is the multiplicity of the focal point.

The twist condition immediately gives you that the set of focal points along the geodesic obtained by projective $\phi_t(p_x)$ on $M$ is discrete.

I don't think that what I will now write is the best or quickest answer to your question, but maybe it is worthwhile to set it down because the point of view is a bit of useful folklore.

First let me translate the problem from Morse theory to Hamiltonian dynamics:

1. Instead of the submanifold $L$ consider its conormal bundle $\nu(L) \subset T^* M$ defined as the set of all covectors bases at some point of $L$ that vanish on the tangent space to $L$. The conormal bundle is a Lagrangian submanifold.

2. Consider the geodesic flow $\phi_t : T^*M \setminus O \rightarrow T^*M \setminus O$ on the slit cotangent bundle. Consider also the flow $$ D\phi_t : T(T^*M \setminus O) \rightarrow T(T^*M \setminus O) $$ obtained as the differential of the geodesic flow.

3. Last, but not least, consider the bundle of tangent Lagrangian planes over $T^*M \setminus O$ that I'll denote by $\lambda(T^*M \setminus O)$.

The flow $D\phi_t$ induces a flow on $\lambda(T^*M \setminus O)$ that I'll denote by the same symbol. This flow has an important "twist" condition: if $p_x$ is a point in $T^* M \setminus O$ and $\phi_t(p_x)$ is the integral curve passing through it, consider the vertical Lagrangian tangent plane $V_{\phi_t(p_x)} \subset T_{\phi_t(p_x)}(T^* M \setminus 0)$ and the curve of Lagrangian planes in the (symplectic) vector space $T_{p_x} (T^*M \setminus 0)$ defined by $$ t \mapsto D\phi_{-t}(V_{\phi_t(p_x)}) =:\Lambda(t). $$

4. The twist condition satisfied by geodesic flows for Riemannian and Finsler metrics (among other non-degenerate Hamiltonians) implies that if $t$ and $t'$ are sufficiently close, then $\Lambda(t)$ and $\Lambda(t')$ are transverse.

5. Now we have to translate focal points into this language: Consider $p_x$ on the conormal bundle of $L$ (at the point $x$) and the orbit $\phi_t(p_x)$ $(t \in \mathbb{R})$.

Proposition. The point $y$ obtained by projecting $\phi_s(p_x)$ onto $M$ is a focal point if the Lagrangian plane $\Lambda(s) \subset T_{p_x}(T^*M \setminus O)$ does not intersect the (Lagrangian) tangent plane of the conormal bundle $\nu(L)$ transversely at this point. The dimension of the intersection is the multiplicity of the focal point.

The twist condition immediately gives you that the set of focal points along the geodesic obtained by projecting $\phi_t(p_x)$ on $M$ is discrete.