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

Totally of the cuff: I believe you can recast the Sims in the discrete Fourier transform as integrals of step functions. So it should work. However the width of the steps in the discrete dirtier transform is fixed for each n. So you'll need to be a bit clever to make sure your steps line up right of you want to build up a single infinite basis. And I would be a bit afraid that the need for things to line up would end up losing some needed span... So at least the are done things which need to be checked.

The sin and cos basis is very convenient. Even more convenient is the e^(i.n.t) basis...

Yes. See Haar Basis on page 4 of http://www.fourierandwavelets.org/FSP_v1.1_2014.pdf