Many connections or few is not the relevant bit but rather that the places the corner region are connected to are all, of themselves, not distant from each other. After all, the relevant bit is the number of adjacent nations and nations are pretty much always contiguous. Except sometimes temporarily when dissected during war, but that is very rarely relevant at the grand strategy level.
You're falling into the trap of thinking map graphics are significant here. "Distance" is not relevant, because a province cannot be connected to a distant province; it's literally contradictory. And that's not semantics, it's the fundamental point. Interconnectedness is what makes "corners"; if there is an area where there are few paths in and out (because the nodes in the area have few connections), it's a corner. If there is a balanced number of connections in all provinces, there are no corners, as corners in a gameplay sense are areas that are significantly (for some subjective measure of significance) less accessible than others (otherwise there's no real meaning to it).
However, if all players can get placed into a location where they are surrounded by provinces that are as interconnected as the other players' territory, in terms of balance it doesn't matter whether each province is connected to all the other provinces or two provinces.
While this is true, it merely illustrates that your extreme case scenario isn't a useful reflection of gameplay. The difference in balance arises from randomness in placement when nodes are not connected in a homogeneous way, and the particular problem we're discussing is the heterogeneity introduced by edges and corners of the map which does not wrap. Because although it's true in the abstract that node connections are not spatial in nature, the map generator and human map makers both incorporate spatial relationships when creating the web of nodes that is technically the playable map.
You're getting stuck on map image. The image isn't relevant except as an abstraction and something pretty to look at. Again, look at Ragnarok Comes. It's a half-wrapped map that is very densely connected that behaves like a full-wrap map
because of that density.
The map as a data structure has no edges or corners (in the cartographic sense, anyway). It's a connected series of nodes. The issue is not how the nodes are shown on a map (which can be wrapped or not, as the map-maker pleases; I yet again point to Ragnarok Comes), it's how densely connected the nodes are, and whether they're uniformly dense in their connections. If they are not uniformly dense, you'll have "corners" in the underlying nodes even if you wrap the map image.
Players are placed randomly in any case, and actual distances vary according to gameplay and happenstance; distances of capitals take secondary importance.
This is false, and tremendously relevant. Players are placed procedurally, not randomly. The more players (and thrones) there are, and the more densely connected a map is, the harder it becomes to place all players at a "reasonable" distance without increasing map size.
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Again, the relevant points are not the "shape" of the map, or whether it conceptually "wraps"; it's just node-to-edge density (province to connections in Dom terms) and shortest paths. If you want densely connected maps, then we need to move away from Dom3 conventions on what "good" map sizes are, because a map with high connectivity sets itself up very differently, in ways that very directly impact gameplay and often outcome, than a map with low connectivity and the same ratio of provinces per player.
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Edit:
Shoo, keep yer fancy-shmancy drama for later. The game hasn't even started yet.
If you won't decide on a nice and fun map soon enough, I'll just go and make one without asking anybody for opinion, like the arbitrary tyrant I am.
I'm fine with a half-wrapped or unwrapped random with 20-25% water by province count, or some nice non-Pymous crafted map.