Then Clara walks in after the GI and somehow her presence corrects everything. How? It's never explained and it doesn't make any sense.
It's not explained, it's shown: she saves the Doctor by warning him every time the Great Intelligence is about to threaten him.
See my issue is when you watch the episodes that show her inserted into. I find a hard time comprehending anything she could've done that effected the outcome of the adventure any differently if she had not been there.
With the 1st Doctor she tells to steal a defective TARDIS. Helpful yes, but I recall in one of the classic era episodes the Doctor saying he stole the specific TARDIS from a museum in some sort of premeditated plan. The episode the "Doctor's Wife" has the TARDIS corroborates the Doctor's story when she says she stole him. I'll search for the episode when the Doctor states his plan to steal a TARDIS later.
The clips of the 2nd and 3rd Doctor are of him running and driving from a teleporation spiral in the episode "The Five Doctors". His 2nd persona runs by Clara, but he is still captured by President Borusa. His 3rd persona drives by Clara and he sees her in his rear view mirror yelling his name but he is also captured by Borusa.
The 4th Doctor clip is of him on Galifrey during the episode "The Invasion of Time" part 2 of 6. The Doctor walks down a hallway and it is Leela not Clara who is pursuing him. The Sontaran invaders don't make an appearance till the very end of episode 4. So there is no immediate threat Clara is saving him from in episode 2. Also it was the Doctor's plan I seem to recall to let the Sontarans penetrate Galifrey's defenses and occupy it.
The clip with the 5th Doctor involved Clara looking at the Doctor while he was trapped in the Matrix of the Time Lords. This during "The Arc of Infinity" part 3 of 4. A place he was put into by a Omega after the Time Lord council tried to execute him. Again Clara does nothing to save the Doctor. It is Omega himself who frees the Doctor and returns hims to Gallifrey for his own ambitions.
The clip with the 6th Doctor has no episode from the Colin Baker era. All we see is Clara in the TARDIS and the 6th Doctor walking behind her.
The clip of the 7th Doctor from the episode "Dragonfire" again has Clara looking at the Doctor but not actually doing anything to help him. The 7th Doctor dangles off the side of a clip holding on to his umbrella. However it was Sabalom Glitz who comes to his rescue.
This link illustrates everything I just detailed out.
http://news.thedoctorwhosite.co.uk/the-name-of-the-doctor-past-references/
Rewatching "The Name of the Doctor", and I still haven't figured out how Clara saves the Doctor. The Great Intelligence walks in to the Doctor's time stream and with his knowledge of the Doctor's life to turn all his victories in to defeats. Then Clara walks in after the GI and somehow her presence corrects everything. How? It's never explained and it doesn't make any sense. Did she just nullify the GI? If so how? Clara lost her memory when she was shattered in to millions of copies. The GI retained his.
Watching the two episodes that featured Clara before she became a companion proper; Asylum of the Daleks and The Snowmen. Clara's presence didn't thwart anything. Remove her from both episodes and there isn't much threat to the Doctor's life that she saves him from. The scenes with Clara edited in to clips of the Doctor's previous lives also don't affirm any contribution of her saving the Doctor.
Er, didn't Clara wipe all memory of the Doctor from the Dalek archives in Asylum? That sounds pretty useful, also she was
providing assistance for the Doctor, Rory and Amy that kept them alive.
As for the Snowman, she saves the Doctor by getting him involved, if she hadn't the GI's plan would have been more likely to succeed, and either the Doctor wouldn't have got inovled, or might have got involved later when there was less time to resolve the matter.
With "Asylum of the Daleks" you have to think about why the Doctor was brought there in the first place. The Daleks wanted him to investigate the strange signal coming from the planet which was Clara "Oswin" Oswald. Yes she deletes all the information about the Doctor in the Dalek database. However if she was never there in the first place, the Doctor would not have been there, and his life wouldn't have been in danger while he was trying to rescue her. I'm not trivializing her act, but pointing out in this situation, she created own issue and resolution.
With "The Snowmen", Clara was instrumental in getting the Doctor to investigate the threat of the snow before it was too late. She didn't really save him from death. She saved him from his depression funk he had been in after losing Amy, Rory, and River. That counts for a lot I think.
As for why her memory is splintered whilst the GI retains his. Firstly what proof is there that the GI retains his anyway, and secondly are you really debating how a creature that's pure thought, hundreds, potentially thousands of years old, composed entirely of information can do something a normal young human female in her 20s can't?
The proof that the GI retained his memories is seen in the episode. Madame Vastra reports star systems are disappearing. Jenny vanishes presumbly dead since the Doctor once saved her life. Strax turns hostile like he doesn't know who Vastra is. The Doctor's act have been negated and or prevented by the GI's meddling in the Doctor timestream.
I'm not debating anything. I'm trying to ascertain how exactly Clara was able to right every wrong the GI imposed on the Doctor's timeline upon entering it. It's easy to digest the GI corrupting the Doctor's timeline with his vast knowledge and previous history with the Doctor spanning back to the Doctor's 2nd incarnation. All we see Clara do is walk into the Doctor's timestream and somehow that miraculously fixes everything.