You seem to have something against the FFT, yet continually compare other transforms to the "sweet spot" for the FFT, ie. sinusoidal and periodic functions. Without trying to write a treatise on the subject (there are many fine books), a few observations.
The primary purpose of any transform pair is to perform a rotation in function space, for the FFT we are familiar with the time domain and frequency domain. The wavelet domain is not so simple. The FFT is very effective for sinusoidal and periodic functions, ie. those that have no localization in the time domain. If you have a sinusoidal input, the infinite signal in the time domain can be reduced to a pair of numbers (one if you take the power spectrum). That is very efficient, other periodic signals can usually be approximated by a small number of terms.
Wavelets on the other hand are chosen to be localized in both the time domain and the wavelet domain. If you look at a periodic signal like your sine wave, you see a mess, and to recreate the input signal you will need to keep an awful lot of the points around for the inverse transform. You may be able to set the last 1/3 to zero, but that's it. To see an effective application, you should be looking at localized functions, like an impulse. Put an impulse into the wavelet transform and you get something that is still a bit complex, but pretty simple. The FFT of an impulse contains components at all frequencies and the wavelet clearly wins in this case.
On paper, there is usually a clear choice, and the FFT almost always wins for spectral estimation. On the other hand, if you are trying to compress "real world" data, for instance an image, then there is a choice to be made. Standard JPEGs are FFT based, you take the FFT and try to keep just the largest frequency components. As you probably notice, edge contrast usually suffers, especially at higher compression. With wavelets, it is typical to have slightly better contrast leading to better preservation of details for a given compression.
In short, my opinion is that transforms are useful when they simplify the problem. Wavelets do not simplify the representation of a sine wave.
