I’m sure there are a few cases where what you’re describing happened, but more often a character where you can choose the sex is intended as a self-insert, and the story is also written with that in mind. This is usually expressed through dialogue choices, branching paths, multiple romance options. With some games those differences are minor and with some they’re major, but they’re almost always there (or, alternatively, you get the Pokemon-style silent protagonist, barebones story, customization is in the Pokemon you use or whatever).

In other words, the fact that the personality is perceived as different depending on the avatar is a feature, not a bug. So, when you say “actually, take it out,” you’re not protecting the story/game as intended—you’re actually damaging it.

Not to mention that even if we say “allowing a self-insert is always strictly worse than what you would produce if you redesigned the story to have a specific protagonist,” what you’re saying would still only apply as writing advice for future projects. A remake of a classic game is based on an already finished product that people loved as it was, and as such its artistic integrity is entirely dependent on not fucking with that product.