Creation of ester from acetylene derivative
No, your approach doesn't give the right product. There's a carbon missing between the benzene ring and the carbonyl carbon. See the difference in the products below.

I don't know exactly what your solutions manual suggested, but I've included a conservative approach through phenylacetaldehyde. As you suggest, it's not absolutely necessary to go through an acid chloride to make an ester from a carboxylic acid. The Fischer esterification will work. Also, for many carboxylic acid to ester conversions, you can simply react the carboxylic acid with thionyl chloride and the desired alcohol all in the reaction together, rendering it a one-step transformation.