Firstly, I do agree that Iroh wasn't terribly involved with the White Lotus until Things Got Real TM. Mostly because, as I see it, the same reasons as Gran Gran to a certain extent: they had something they considered more important. Gran Gran had her village, her son, and her grandchildren, on top of all the other daily tasks that any small village has (less people, more work). Iroh had his nephew, who he was trying to keep alive. Remember, it wasn't Ozai who got the boat, who found the crew... It was Iroh.
And could the White Lotus have recognized that there were differences? Of course they could have... But again, as I stated... They were an isolated people: it's heavily implied that while there might have been other villages in the Southern Water Tribe, they weren't a nation as such; at best, much like many more tribalistic societies, they were a group of people with a similar culture and values that occasionally interacted.
Now, the next part takes putting some puzzle pieces together, admittedly. However, at one point, it's stated that nobody came to help the Southern Water Tribe while they lost their benders, one by one in these raids.
We can also see that there's no technological 'drift' per say from their nearby (as a relative term for people dwelling at the south pole) neighbors. One of whom was the Earth Nation, which was relatively advanced for the world/setting. To a point where they have next to nothing beyond a basic dark ages village. Even their ships were small, little more than Scandinavian karves.
Almost every single person the Gaang meets is shocked to hear that Sokka and Katara are from the Southern Water Tribe. Several of the graphic novels state repeatedly that the Southern Water Tribe was near extinction
Given all of these facts, it is highly unlikely that the Southern Water Tribe was interacting with anyone at all outside of scattered meetings with neighboring tribes.
Now, moving further into Kanna not being 'alone': you're right, she wasn't. But again, even discounting the fact that we have no evidence they interacted with anyone outside of their tribe, and the numerous pieces of evidence saying that they didn't... Her leaving on 'business trips' would a) be highly unusual, b) a detriment considering they were missing almost an entire generation, and c) incredibly inconvenient to a group of tribal villages.
Now point b first: as we can see in the show, and as we're told through various characters' pov (both in the show and the books), the Fire Nation kidnapped any water bender they could find. Given that about 20-25% of any given 'nationality' were Benders, this would have been devastating to a population that was already much smaller at the start of the hundred years war, and after even fifty years of raids, this would've lead to a massive generational impact. For reference, look what happened in WWI when the British put Irish soldiers on the front lines, and how that decimated the Irish population.
This means that the older generation would've been needed to teach the few remaining grandchildren of each successive generation. Elderly people would've been incredibly important, because as more and more people were taken, there was a greater and greater chance for lost knowledge.
As for point c: again, as we can see... They had next to zero technological advances. They were barely Iron Age tech, much less anything further. In a society like that, even children often have basic tasks to do around the village, and as we can see from both Katara and Sokka's upbringing, they weren't 'children' as we would think of a child. Hell, we meet the two of them while they're out fishing for food for the village, at fifteen and fourteen respectively.
So yes, one person leaving -much less a valued elder with all the knowledge they possessed -a tribe for months at a time would've been inconvenient, if not outright devastating.
And it bears reiterating: they had a Water Tribe member. For all Pakku's faults... he was Water Tribe. If he even bothered to share that there was another group of Water Tribe people to the south -if he even knew they still existed -as far as he was concerned... it was the same tribe. He tells Katara a few times that 'clearly' the Southern Water Tribe has lost its way if she's doing this or that, and everyone in the Northern Water Tribes seems quite surprised at everything she does. So clearly they were unaware of the cultural/societal shift that had happened in the first place.
So unless Gran Gran had already known about the White Lotus, and went out specifically to hunt them down... Why would any of them bothered to go looking for someone in a group of people thought to be on the brink of extinction, or extinct, who were supposedly the exact same as this other group of people who are over there?
To put it in context... This would be like insisting that they wanted/needed people from each city in the Earth Kingdom. Well... why? Why bother? They're more or less the same people, and we can't get one person from every village that maybe does things slightly differently (again, remembering that the Northern Water Tribe was apparently unaware of the massive cultural/societal split that had occurred between them and their sister tribe).
And as to my last point, which I believe was your third point... The White Lotus wasn't dedicated to 'teaching' the Avatar, or 'protecting' the Avatar. Remember, as far as the world was concerned, the Avatar was dead. And as far as the world was concerned... The Southern Water Tribe was either also dead, or mostly dead.
And even if they hadn't thought that... Gran Gran wasn't a Bender. She would have had nothing of value for the Avatar, beyond things that he could've gotten from any halfway decent human being, or any other old person in any other random place.
And as a final point... Each member of the White Lotus was incredibly talented in one particular area; maybe not Bending, but they were either leaders of their people, had renown or glory attached to their names, or were people that had been/were very highly placed in their respective culture/society.
Gran Gran... was none of these things. She had been married to the chief (presumably, given that Hakkoda is chief now and it seems to follow hereditary succession rules), but she wasn't the chief. She wasn't the possessor of unique skills; she was no more or less competent than any other elderly woman of the Southern Water Tribe. Her only minor claim to something approaching 'fame' would be that she almost married Pakku a few decades ago... but didn't.
Again, could every single one of these issues have been possibly written around? Could the writers have simply 'done it', and ignored everything about their own world/creation? Sure, they could have. But given the knowledge that we have of the world and the characters, it would have been a major plot contrivance to have her be a member of the White Lotus.
Especially when you consider if she had been, at some point you think she would've mentioned, "Hey, the Fire Nation is wiping out our people, and we're literally living in small igloos and sailing in boats slightly larger than the average bison; wanna help us with that?"
Having Gran Gran as a member of the White Lotus would have ruined the entirety of the Southern Water Tribe story line, ruined Gran Gran's character, or ruined the entirety of the White Lotus purpose. Because we'd have to assume that...
A) She never said anything, and let her people be decimated and nearly driven to extinction, including making everyone's lives far more complicated by not getting any of the other nations' technological advances brought down, because she just... didn't care at all about the quality of life for the Southern Water Tribe, who were barely surviving.
B) She did say something, and the White Lotus -whose purpose was the preservation of knowledge and culture -simply laughed and said, "Haha, sucks to suck, loser!" and did nothing to stop this culture from being extinct.
C) She did say something, and the White Lotus did help, but through magical plot devices, somehow they were nearly driven to extinction anyway.
So we have both in-verse reasons, meta reasons, and plot reasons for why it wasn't feasible to include her in the White Lotus.