Assuming I need to get a large cargo or a lot of people safely to Earth's surface in one go in a single-use capsule of the Apollo CM / Soyuz descent type but larger, and assuming that the capsule is already in space (so launching it probably isn't an issue), what might limit how large the capsule design can be? Would there be any particular advantages to a larger capsule, again assuming that launch isn't a consideration?
Let us consider the square cube ratio. If we enlarge an Apollo CM capsule by a linear factor of 2, its surface scales with factor 4 (the square of 2) and its mass and volume scales with the factor 8 (the cube of 2).
But for the hot phase of reentry we need the same path length in the upper atmosphere. So the deceleration (g force) should be the same as for the original capsule size. To decelerate 8 times the mass, we need 8 times the drag force. But drag force scales by the square of the diameter, not by the cube. So we need a longer reentry distance to slow down, but the height of the atmosphere is still the same as for the original Apollo capsule.
If the cone shape of the capsule should be the same, we need a capsule with the double diameter and height, 8 times the volume but only 4 times the mass of the original Apollo capsule. But this would require an extreme lightweight construction and would limit the possible payload too much.
If we would manage to slow down to the speed for a parachute deploy at the necessary height, we have another problem. When we use 3 parachutes with the double diameter, the weight of the capsule should be only four times but not eight times the original weight. So we would need 6 instead of 3 parachutes with the double diameter. Or 3 parachutes with 2.828 times the diameter (the square root of 8).
But both sets of parachutes will have more than 8 times the mass due to the larger area and mechanical stress. So for a very large capsule the necessary parachutes will be too heavy.
@Uwe's answer is great, and presumably to the question
Is there a practical maximum size for a space capsule returning to Earth?
the answer there is pretty much "yes" based on current capsule technology. Space planes and "rockets with flippers" can be larger perhaps, but @Uwe's square-cube argument would still apply.
I'll address the last bit:
Would there be any particular advantages to a larger capsule, again assuming that launch isn't a consideration?
and my answer is "not in terms of ease of reentry", and for the same square-cube reasons. The smaller reentry capsules for automated sample return don't need to push things to the same extreme limit in terms of heat shield and parachutes, and the lower mass allows for mid-air retrieval.