Because if we crown a woman who's not a family member and not the Queen of the Seven Kingdoms—and is known to be a very close confidant of Princess Rhaenyra—as the Queen of Love and Beauty, and then reveal that we are Princess Rhaenyra, we come that much closer to publicly outing both Alicent and ourself as lesbians.
No one has any reason to cast aspersions on the honor of this specific Silver Falcon right now, and the idea that we could spawn copycats who might sully our reputation and Alicent's by proxy is (I think) a much less immediate danger than turning to the crowd and saying, "By the way, I'm the royal princess and I think my close friend and handmaiden is the most beautiful woman in the world".
This has already been addressed
Alicent knows that Rhaenyra is going to unmask herself upon winning because that was always the plan and she wanted to crowned Queen of Love and Beauty anyway
Explicitly, they openly communicated with each other about it
Alicent os also reasonably sure she can pass it off as an innocent gesture of playful friendship since Rhaenyra obviously can't crown a man
Alicent and Rhaenyra have already talked about it, and she's okay with it
More than okay, actually
Alicent actively wanted to be crowned the queen of love and beauty as a big gesture of their knightly romance
And she's reasonably confident she can provide an excuse to onlookers that obviously Rhaenyra couldn't crown anyone else, so her "close friend" and handmaid was the only reasonable option available
"When I win," you assured her, "you'll be crowned the Queen of love and beauty."
Alicent's face turned an adorable shade of red.
"You're fine with that, right?" you asked as you placed a hand on her shoulder. "It is a very open gesture."
"I don't think it is," Alicent replied. "When you win and you reveal yourself, if anyone asks, it will be easy to explain there was no way you could give the crown to any man who had your affections. That might be taken as a serious insult. So instead you gave it to your good friend and handmaiden."
Alicent had made clear from the start of their relationship that she wants Rhaenyra to crown her if she wins a tourney.
There is going to be rumors about the two of us anyway
But I will be frank, that was always inevitable
There are already rumors of that sort about Rhaenyra, and they've probably existed since the moment she started getting into fist fights with stable boys over how they treated the maids, and took up a training sword in the field and dressed herself in armor
And if they didn't then they certainly did the second she danced with Jeyne
Alicent is smart enough to know this
It's just important to her that she and Rhaenyra maintain a plausible cover of deniability because she has a specific thing about staying inside social boundaries and expectations
A view of hers that is beginning to change, I will note
She's literally more likely to get caught off guard by us asking to ride off with her than she is us unmasking ourselves
And if we ride off we'll have to do this all over again at highgarden
At which point we either claim the Falcon identity again, and all your worries about outing Alicent return tenfold because the Falcon is known to have rode off with her in a big dramatic romantic gesture
Or we come up with a new mystery knight in which case all this was for nothing