First of all, yes, the manufacturing process absolutely dictates constraints on cell thickness for both mono- and multi-crystalline (mc) Si solar cells. For mc-Si cells, thicknesses below 90um increase wafer breakage in the line significantly which reduces yield.
To understand the thickness requirements from the solar cell physics perspective, you need to consider that electron-hole pairs are 'generated' at different thicknesses in the material and need to diffuse (travel) to the depletion layer while avoiding recombination on the way.
Anyway, I assume that the relevant thickness for the first point (absorption) is the thickness of the depletion region rather than the total thickness.
Not really. The relevant thickness for absorption should be understood differently when talking about the emitter regions (n++-doped) and the base region (p-doped).
In the emitter region, where most of the light is absorbed, the concentration of electrons is already very high due to the n++ doping. The high doping is also necessary to allow good metal contacting on the front side. After light is absorbed, an electron-hole pair is 'generated'. Here, holes are minority carriers and need to diffuse to the depletion layer so they can be collected. Therefore, the thickness of the emitter region needs to be small enough (< 1um) in order for the holes to have a chance to 'survive' and not recombine with the surrounding electrons before reaching the depletion layer. The rule is to have the emitter thickness within one diffusion length of the holes. Hole diffusion length itself depends on the material's crystalline quality but MOSTLY on the level of majority carrier concentration (doping).
In the base region, the light wavelengths that will be captured there generate electron-hole pairs which are essential to increasing the current output of the solar cell. Here, electrons are minority carriers and need to diffuse to the depletion layer and avoid recombining with majority carriers on the way. Because the doping level is much lower than in the emitter, having a small thickness becomes less critical as electrons can diffuse for several diffusion lengths. The electron diffusion length here depends mostly on material quality (defect concentration, grain boundaries, dislocations...).
This suggests that, given a total thickness of the p-n junction, one should try to maximize the thickness of the depletion region and minimize that of the remaining p- and n-doped regions
Yes, however, the main way to increase the size of the depletion region is to increase the n++ to p doping differential which in turn increases the minority carrier recombination rates in the emitter and base areas.
For the layer facing the incident light, that would have the added benefit of reduced absorption in the conducting part
You wouldn't want to reduce absorption in the emitter area or anywhere really. Absorption of light anywhere in the material means 'generation' of electron-hole pairs which is something you should try to maximize.