Yes, we will find the optimal solution by checking every corner of the feasible region. The problem is exactly the computational complexity.
Suppose that the feasible region is given by a system of equations $Ax = b$ with $x\ge 0$, where $x \in \mathbb R^{100}$, $b \in \mathbb R^{50}$, and $A$ is a $50 \times 100$ matrix. Then to check every corner of the feasible region, we should try all $\binom{100}{50}= 100\,891\,344\,545\,564\,193\,334\,812\,497\,256$ choices of $50$ basic variables: first to see if they produce a feasible corner point (that is, all $50$ of those variables are nonnegative when we solve for them, setting the rest to $0$) and second to find the objective value so that we can compare.
This brute-force search will take far too long: if we check a billion points a second, we'll be done after approximately $200$ times the age of the universe so far. On the other hand, the typical linear program of this size can be solved nearly-instantly with better algorithms.
For any particular pivoting rule, there are examples like the Klee–Minty cube where the simplex method is also exponentially slow (in other words, just as bad as checking every corner). But this is very rare, and there are many results showing in various ways that on average, the simplex method will be much faster. And there are other methods with a guaranteed polynomial running time.
