[_-__---]

They can[0]; however, this requires some further construction.

One major benefit of the sin/cos basis is that each basis function maps uniquely to a dirac delta in frequency space. This allows for power/energy symmetries between time and frequency domain representations of a signal and allows for more tenable frequency filtering, e.g. removing 60/50Hz noise or isolating a particular frequency band.

A haar wavelet or modified square wave basis may provide you with a simple orthonormal signal basis, but each wavelet has a frequency representation with infinite support in frequency space. This is (a) untenable and (b) eliminates the possibility of frequency-specific filtering. Wavelet analysis is more useful in specific cases.

[0] https://en.wikipedia.org/wiki/Haar_wavelet

[alok-g]

When you use the term "frequency space", you are already thinking in terms of the frequency of sin/cos functions. If you actually think in terms of frequency in a different orthonormal basis, you'll find that the dirac delta characteristic you mention is not specific to sin/cos, thereby invalidating your argument.