The number of board states is on the order of 10^45. That's within a few orders of magnitude of the number of water molecules in the Earth's oceans. So solving chess by exhaustively searching the possible board states is not currently considered computationally feasible.
Instead you're proposing Monte Carlo Tree Search. In this approach, you would use random roll-outs (i.e. playing a game to the end by making random moves, then seeing if you won). This is equivalent to randomly choosing a branch of the possible future board states and seeing if it results in a win, and from that, making moves that you estimate to be likely to lead to victory.
For this to work, given the scale of the possible board states, you need a sampling algorithm that is very thorough and very fast, and most importantly, very representative. Unfortunately, the surface you're trying to optimize over is not very smooth: even the best board position can be turned into a loss with a couple bonehead moves, and if you're sampling random future board states (the hallmark of Monte Carlo, vs. exhaustive, search methods), you have no way to know whether what you have sampled is representative of the entire branch, or if you just picked a rollout that included some really bad plays (or conversely, one where your opponent did something dumb that gave you a needlessly rosy view of your odds).
That's the entire point of the evaluation function: you estimate the value of a set of intermediate board states, which you can more thoroughly explore within a reasonable future-turn horizon. (Until you reach the endgame, where you can realistically do exhaustive search.)
Note that MCTS is in a sense learning an evaluation function: instead of an explicitly stated value for a board state, you're keeping statistics on the number of times a sequence of random moves from that board state have resulted in wins/losses. The problem is, because of how distant the eventual win/loss is from most board states, and because of your inevitable computational limits, you're learning a pretty bad evaluation function.
If your sampling rate was high enough, this could work; but with no evaluation function (or way to approximate an evaluation function, like a neural network), you're going to spend far too much of your time evaluating obviously bad plays and not learning anything actionable. It just isn't computationally feasible given the numbers involved.