If you use a detergent, its molecules would dissolve in the lipid bilayer. As the [D] increases, you reach the point where the membrane can no longer dissolve the detergent, and detergent/lipid mixed micelles form, which may contain membrane proteins. These are small enough not to scatter visible light, thus the solution becomes clear. They are also not pelleted by centrifugation at 100000 g for 1 h, the operational criterium for solubilisation. Digitonin is a special case in that it interacts with the plasma membrane, but far less with internal membrane due to different cholesterol content.

During osmotic lysis the cells rupture, but the internal membrane systems stay intact and can be purified by fractionated centrifugation and/or density gradient centrifugation. Btw, for this step you'd use the buffer only. The sucrose (or other osmotically active agent) are added once lysis has happened, perhaps aided by a few strokes of a glas/teflon (Potter) homogeniser.

I have described membranes and the action of detergents in chapter 15 of ISBN 978-1-4419-7251-4 , and the hypotonic lysis followed by plasma membrane isolation of cell culture cells in doi:10.1016/S0003-2697(02)00639-5.

Very interesting, thank you for the explanation!

Nicely put forth, with detailed explanation @Engelbert Bauxbam.