The question says it all. In typical (wavelet-like) decomposition of a signal, why is only the low pass component chosen for successive decomposition ?
Wavelets decomposition separates out the details/fluctuations/high-pass information from the image or signal. At each step details are separated out from the remaining of the signal. The processing is therefore only applied on the coarse part not the part that already has the details.
A filter bank is just one specific implementation of the wavelet transform. Typically wavelets are motivated and defined differently, namely as a family of (orthogonal) basis functions that are generated from a mother wavelet by translation and time scaling. It turns out that such a family can only be an orthogonal basis if the time-frequency plane is divided into a so called dyadic grid, pictured in this link. The resulting basis transform is then called the discrete wavelet transform.
This specific discrete wavelet transform comes with a fast algorithm, realised by recursive high and low pass filtering and downsampling. The first high pass filter creates the top most row of coefficients in the dyadic grid. The low pass component still contains the full information required for all the other rows. Taking the low pass component through the high pass filter again generates just the next row, and so on.
So it's helpful to not think of the filter bank as the defining property of the wavelet transform but just as a fast algorithm that returns the right result.
That said, there are approaches to adapt the time-frequency decomposition to the signal content, grouping and subdividing filter bank bands as it appears useful. Mathematically, those approaches cannot be understood in terms of wavelets however.