The purpose of the footprint is to ensure a good solder joint, primarily by controlling the exposed copper and the paste volume. As you note, adding a solid pour underneath of conventional non-Solder Mask Defined pads (where the opening in the solder mask is larger than the copper pad) makes them Solder Mask Defined, and in the process it enlarges the exposed copper by the amount of the solder mask expansion. That means a larger fillet, so theoretically a slightly larger solder volume is required, but that's probably a non-issue for most larger parts like these powerSO8 transistors, especially if you're adding window panes for the paste anyway. On smaller pads the change would be more significant, due to the higher perimeter-to-area ratio, but you'd have to calculate the discrepancy to determine where it would require compensating for.
If you expect the device to be connected to large pours, you can create the footprint with SMD pads to begin with, and then avoid the issue entirely. In the powerSO8 case, you could just create a rectangular pad for each of the three source, drain, and gate that is at least as large as your current mask openings, void the standard mask expansion (in Altium, set a negative manual expansion value), and then draw the desired exposed copper shapes (roughly the size of your current pads) in the mask layer.
SMD pads can be challenging on fine pitch devices, since they necessarily reduce the clearance to adjacent pads, so recommended footprints for some devices will show both pad types, like the attached footprint drawing for the TPS25751. A potential downside to combining pad types is that any registration error between the copper and the mask layer becomes an offset in the exposed copper for the SMD vs non-SMD pads, so you'd have to confirm that the fabrication tolerances are tight enough -- I suspect that's not much of an issue for any reputable fabricator, but YMMV.
In terms of the benefit of the additional copper, you can calculate that, or just approximate by looking at the number of squares along the path. Looking at your screenshots, the left option has the equivalent of 1/2 square between the central pad area and the pour above, while the right option has the equivalent 1/6, so has about 1/3 the thermal/electrical resistance in that top layer connection. The actual difference is probably a bit less, but depends on the part package (exposed bottom pad area, leadframe thickness, die position, etc), plus you have the vias to lower layers to consider, which may matter more or less depending on how the heat and current flow in the surrounding parts of the system.