Link has always been a character who walked a tightrope between being a character in his own right and being a player avatar. In that regard he is very much like the protagonists of many of the early Dragon Quest main characters or even the Avatar from the Ultima series of games. These are games from before detailed character creation took off in videogames, but they still embraced the idea that the main controllable character is supposed to be a link between the player and the game world.
You can see that desire to maintain Link as a player self-insert character in a lot of key game decisions:
1) In every Zelda game up until now, you can name Link whatever you want. In the multiplayer games in particular, there is no difference made between Link's name and your player name. "Link" is not even used as a default; if don't read the manuals, you would never know that it was supposed to be his name.
2) Link never speaks. To have him speak would be putting words in the mouth of the player, in essence. The only time Link ever says anything, it is only by giving the player a choice of different dialogue options. Some of the DS Zelda games go even further by asking the player to use the microphone in order to speak for Link in a literal manner.
3) Link has often been depicted as an everyman character: a normal person like anyone else who steps up to become a hero. This isn't an uncommon role for protagonists that the player is supposed to relate with.
However, over the years, Link has become established more and more as a particular character. The name Link has become much more strongly associated with the character thanks to appearances in other games, such as the Super Smash Bros. series. More significantly, starting in Ocarina of Time and even more-so in Twilight Princess and Skyward Sword, Link has been given ever more complex relationships, emotional reactions, and story details surrounding him. Furthermore, the Zelda series has increasingly embraced the concept of the timeline, and it has thus created a more specific story role for Link outside of being an everyman character. Link is now a very recognizable character in how own right, with a long list of recognizable character traits.
Now, if Link is supposed to be a player avatar, then it only makes sense for the player to be able to customize the character. That is after all the current standard. However, if Link is a character in his own right, then it makes sense to leave him be. The fact that he exists in an awkward middle-ground is why there is so much debate and high emotions over this topic.
Now, if they had introduced a gender option sometime around when A Link to the Past was released, or shortly after it, I don't think there would have been as big of a debate over this topic. It really was starting in Ocarina of Time that Link started to be a more developed character. Now that it has been almost 20 years since then, forgetting about how Link has been so strongly established as an iconic character is difficult. Switching from a well-established character to a customizable player avatar is a big change. However, Link remains a player avatar figure, and I fully understand and empathize with people wanting their player avatar character to better resemble them.
Right now, I think the best option is to leave Link as he is, and include the option to play as a different character such as Zelda or a more purpose-built female hero. I think that the awkward balance between established character and player avatar has become what people expect from a Zelda protagonist, but that doesn't mean that we can't have options.
As an aside, I don't think lore arguments, no matter whether they are for or against Link being able to be a girl, are worth anything. Lore is constructed in service to other goals, and the main reasons people are for or against Link being female are meta reasons. So arguing about whether or not it makes sense for Link to be a girl on the basis of the timeline or reincarnations or stuff is fruitless.
