Does it make sense to use a wavelet that is equal to a sine of one period?
With all due caution, no in both cases (title and body question). I'll start with the second one. 


Continuous wavelets use all dilations of the mother wavelet, which are not accessed with the STFT
The STFT is complex in general, and the windowed sine is not.


For the first one: I  never tried it, and do not remember having seen it in use, and one should check first whether it satisfies the admissibility condition (first thought: seems OK on 0 in Fourier, and decays at infinity, I'd expect families of synthesis wavelets). You can compute the effects in Fourier using SE.Math: Fourier transform of 1 cycle of sine wave.

However, the sine falls abruptly at its edges, without differentiability, in other words it is not smooth enough. So I would not expect a  more interesting interpretation  of the scalogram than with other continuous wavelet.

But it could be used as an exploratory tool (analysis) for some specific signals. You can think about  hybrid methods as well: try to fit (in more or less robust ways) a parametric sine model, at different scales.