The Claisen reaction happens under equilibrium conditions, and the addition of the ester enolate is highly reversible. Your mechanism omits the very important end of this reaction, which is that the β-keto ester product is highly acidic and easily deprotonated to yield the corresponding enolate.
The enolate formation drives the reaction forward according to Le Châtelier's principle. Very importantly, your question about the α-hydrogen. The deprotonation of the β-keto ester is not possible without two hydrogens on the original ester that will serve as the nucleophile.