During the first episode, it's 100% exploration, because all our Q values are 0. Suppose we have 1000 time steps, and it's terminated by meeting a reward. So, after the first episode, why can't we make it 100% exploitation? Why do we still need exploration?
You can't guarantee that you have taken every action from every state, even with 1000 time steps. There would be multiple outcomes:
The episode terminates, either by success or failure before the 1000 time steps. The agent is trying to maximise reward, if this is achieved by taking less than 1000 steps then it will do. It won't just walk around until it hits an arbitrary number of time steps.
If you have more states than time steps then you will never be unable to visit all states and so you cannot guarantee that the policy you followed was optimal (and hence would still want to explore). Even if you have #states = #time-steps then you will almost certaintly have more state-action pairs than timesteps. The only time this would be equal is if from every state there is only one action, which would be a trivial problem that wouldn't need RL to solve.
BlueTurtle's answer is good, but I'd like to add something.
Your question realistically has nothing to do with Q Learning, in fact, you can ask the same thing about just about any RL algorithm. In fact, even in multi-armed bandits, you can easily see why your proposed method is suboptimal (please don't interpret this as a criticism, because I think your question is a very natural one). My suggestion to you is to read up on multi-armed bandits since they're much simpler to analyze. I think even the Sutton and Barto book deals with your proposed method explicitly, and mathematically proves that other strategies are better.