This is a common question, but notice your reasoning is this:
Black's king is allowed to move into check because white's king isn't.
Sure you could come up with alternative rules like if the kings are captured on successive turns then it's a draw, but you can't argue Kxg7 is legal or that it's illegality is illogical.
True, but the white knight is pinned. The semantics are not as simple. Can the king really be considering "in check" because check suggests danger though the white knight cannot move.
True, but the white knight is pinned.
What does it mean to be pinned? Why can't the knight move?
You can also think of it like this: whose king would be captured first?
I am clear on why the rule isn't changed, it's a discussion about why it should/shouldn't be or how often this has occurred for others.
I'd say this topic comes up about 2 times a month 
I understand the idea. The idea is a piece pinned to a king is sort of disabled, but that's not really the rules it's more of a shortcut way of thinking so we don't waste energy calculating things like (if it were white to move) Nf5 RxK. We just tell ourselves the knight is a non-piece.
As for the effect on games and tactics, I'm not sure. Making stalemate a win is another common topic, and that affects endgame theory enormously (and chess as a whole) but this one... off the top of my head I don't think it changes much.
Think of it this way. Capturing the King ends the game. It does not matter that the knight is pinned because the game is over when the black king is gone.
So, yeah, it's an interesting idea. I haven't thought about how it might change openings or endgames.
Since I think the only positions affected are ones where the king is attacking a piece, I'd guess it'd mostly change endgame theory, and maybe some big sacrificial attacks when there are a lot of enemy pieces near the king. But I can't think of any theoretical position in particular where it'd make a difference but someone could probably come up with something.
Even if 1...Kxg7 is possible white would still win. e.g. 2.Kd6+ Kf7 3.Rf2+ Ke8 4.Rf8#
With this proposed new rule, from your position Black plays Kxg7... now white plays Rb7 , pinning the R to the Kg7.. after Kf7, then you are saying Ng5 ch is ok... ( the W king is not in check as the Black R is pinned ) or if Black plays Rxb7, now the unpinned knight is free to "take" the K?
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You suggest this rule as an alternative to the current one (which you claim is complicated) but look how difficult it is to make it work!
You forget that the object of the game is to capture the king. If he tool the pawn with his king, then the night would take the king and win the game.
Today I played a game with my brother, and a key endgame position arose in which I saw a move for him as black which was illegal, but in my opinion, it shouldn't be.
With black to move, he cannot legally capture the white pawn. However, why would the black king be in any danger if the white knight is pinned? If allowed to capture, this position would be theoretically drawn instead of winning for white.
The proposed new rule is allowing a king to move into check if the king cannot legally be "captured" by the piece attacking the square because the defending piece is pinned to its own king.
Some interesting variations arise after this, like the possibility that the white king could give check to the black king simply by unpinning himself from the knight.
Should this proposed rule be allowed? What are the implications for engines, tournaments, and overall endgame techniques? Does this present a whole new slew of tactics? I wanted to consult the community to see if anyone has noticed this before.