Advantages of using Bellman equations

Question

Suppose we wish to solve a simple infinite horizon cake eating problem, such that: $$ \max_{\left\{ c_{t}\right\} _{t=0}^{\infty}}\sum_{t}^{\infty}\beta^{t}u\left(c_{t}\right) $$ subject to: $$ c_{t}+a_{t+1}=a_{t} $$ where $a_{t}$ is the total amount of the cake left over for tomorrow. We can use Lagrangian techniques, obtain the Euler equations, and pin down the optimal levels of $c^{*}.$ Alternatively, we can also write this as a Bellman equation as: $$ V\left(a\right)=\max_{a'}\left\{ u\left(a-a'\right)+\beta V\left(a'\right)\right\} $$ and then either solve it numerically (or even obtain an analytical solution in this case, by guess and verify and matching coefficients). My question is this: what are the advantages of using the Bellman approach? The textbook answer is: it helps breaks down an infinite period problem into sequences of two period problems. But, it is still not obvious to me how this is in fact helpful, preferable over simple Lagrangian based optimality conditions. What am I missing?