Why isn't it wise for us to completely erase our old Q value and replace it with the calculated Q value? Why can't we forget the learning rate and temporal difference?
Here's the update formula.
Removing the learning rate will likely yield poor convergence to the optimal policy and optimal Q-values. Note that the current policy is completely dependent on the Q-values, as we take the action with highest Q-value in a given state (with a few other considerations such as exploration, etc.). If we were to remove the learning rate, then we are making a relatively large change to our Q-values and possibly to our policy as well after only a single update. For example, if the sample rewards have great variance (e.g. in stochastic environments), then drastic updates to a single Q-value may occur simply by chance when a learning rate is not used. Due to the recursive definition of Q-values, a few poor updates can undo the work of many previous updates. If this phenomenon were to occur frequently, then the policy may take a long time to converge to the optimal policy, if at all.
Underlying the temporal-difference update and many other reinforcement learning updates is the notion of policy iteration in which the estimated value function is updated to match the true value function of the current policy and the current policy is updated to be greedy with respect to the estimated value function. This process proceeds iteratively and gradually until convergence to the optimal policy and optimal value function is achieved. Gradual changes such as setting a small learning rate (e.g. $\alpha = 0.1$) aim to speed up convergence by lessening the frequency of the phenomenon in the above paragraph. Sutton and Barto make comments on convergence throughout their book, with the remarks surrounding line 2.7 in Section 2.5 providing a summary.