Turing machine with semi infinite tape - Prove by construction
You are right. It is possible to shift the contents of the tape when needed. 

The two track solution enlarges the alphabet $\Sigma$ to at least an alphabet $\Sigma\times \Sigma$ to store two symbols on each tape cell. That is quadratic. Your solution perhaps is more efficient in the number of symbols, since we merely have to move them into a new position.

A third proof might be to store the left and right cells of the tape (from an imaginary "middle") in the even and odd cells. That means the simulation would usually fins the next cell in two steps rather than one.

It is not always clear how to measure elegance of proofs :) sometimes a solution comes from a certain intuition, making it appealing.