I think its the consensus is that Great Gospel rainwater actually purges the Jenova cells from your body
Wouldn't that make Cloud lose his SOLDIER super strength and stuff?
Short Answer: No, it wouldn't
Long Answer:
Jenova's most important ability (to the Shin-Ra scientists) is her ability to make "copies" of herself out of living things by "coping" parts of her DNA over their DNA. This is done with her cells (J-cells). Sometime back in the past she looses the ability to do this. (I personally think it's when she was sealed by the Cetra.) It appears that J-cells are still able to overwrite DNA when they are mixed with mako, but Jenova is not controlling them. Instead, J-cells do this because it is what they are programed to do.
This is essentially what happened with the S Project, G Project, and the SOLDIER organization; the J-cells overwrote people's normal DNA to make mutated
human cells. It is the mutations, rather than the J-cells that give Angeal and Genesis their own "copying" abilities and give Sephiroth and the rest of SOLDIER their physical prowess. It goes without saying that the mako also increases their raw abilities.
There is a large amount of evidence that the more J-cells and mako a person is exposed to, the more of their human DNA is overwritten with Jenova's DNA. This would explain why Sephiroth and Cloud are at a much higher level then the rest of SOLDIER. They have the most J-cells and the most mako exposure.
Normally, J-cells don't do more than overwrite DNA because Jenova herself can not control them. Without someone's will behind them, they are neutral. Once the Nibelheim Incedent takes place, this is not the case. Sometime between then and the start of FFVII, Sephiroth takes control of both his mutated
human cells (S-cells) and the J-cells. For just about all SOLDIERs, except Cloud, this is not a big deal because (comparatively) they have hardly any J-cells for him to control them with. They also don't have amnesia. Cloud, on the other hand, has J-cell amounts that are in the same league as Sephiroth's and he has a large amount of S-cells as well. He also has amnesia that Sephiroth is all too willing to use to his own advantage. All of this is why Sephiroth can control Cloud in FFVII.
As it has been stated, Geostigma occurs when the body overextends itself because it is fighting off J-cells. It is significant that the way to contract Geostigma is to get a dose of a J-cells/mako mixture. This sets up the whole J-cells overwriting human DNA thing again, except this time, Sephiroth has control over how the human DNA is overwritten. There is also a lot less mako involved, but that might/might not be significant.
All that said, what Aerith is doing is getting rid of the J-cells. What she cannot get rid of is all the Jenova DNA the J-cells have "copied over" the human DNA. It is also highly unlikely she got rid of Clouds S-cells because those are really mutated
human cells. In other words, at the end of ACC, Cloud does not have any J-cells in him, but he does have all the traits the J-cells copied over into
his own cells as well as any traits he gets from the S-cells.