Don’t worry about the opening. Clear your mind of any necessities of having anything in the beginning of the game. Do you, and do what feels right in the story. Don’t force selections in or it’ll end up feeling janky or shovelled in, as you said. Honestly, your imagination is the limit with how you should define any main character in your story.
If you wanted to make a character selection screen, you can do that and make it an interactive looping set of choices that people have the freedom to go change all the way up until they make the selection to continue into the story. Make it clear that this is where they set up a character, don’t try to half-hide it behind story elements if you go down this route.
If you want to gradually shape a character, then you can. Absolutely. Selecting the main character’s gender doesn’t need to matter until you make it matter. It can be in Chapter 1, it can be in Chapter 7. The timing is entirely up to you.
Likewise the amount of character definition is up to you. If you feel like gender or sexual orientation don’t come into play, don’t give them an option to select one. Make the story vague enough that whoever reads it can fill in that blanks, play around with expectations. When done well, the reader will never even notice they didn’t pick it, they just filled it in without being confronted about it.
If you want to flesh out the world and other characters in the story, without fleshing out the main character yet, you can jump into the perspective of another person for a little bit. Show what is happening, then find a way to shift the perspective to the main character.