From the Architect's speech (transcribed here), this line from the Architect may be relevant, following shortly after Neo's realization that "the problem is choice":
As I was saying, she stumbled upon a solution whereby nearly 99% of
all test subjects accepted the program, as long as they were given a
choice, even if they were only aware of the choice at a near
So, I'd speculate that the people in the Matrix were aware on a "near unconscious level" of information about the Matrix they hadn't obtained through normal sensory channels (like the option to escape to Zion, which most people couldn't have known about directly through anything they saw or heard), perhaps via some sort of subconscious leak of information through the implants that connected them to the Matrix, or perhaps via some mysterious psychic awareness. Either way, if this is correct, then it could be used to explain why the choice had to be a genuine one, with no "cheats" like withholding information that would be relevant to the One's choice--if there was such a cheat, people throughout the Matrix could become subconsciously aware of it via the same non-sensory channel, and would thus feel humanity hadn't been given a true choice and would rebel against the Matrix more (as they did with the first, Utopian Matrix the Architect mentions).
Another issue may be that it was part of the Architect's nature to be scrupulously honest, even when it may have gone against what he believed to be his own interests--consider this scene from the ending of Matrix: Revolutions:
The Architect: Just how long do you think this peace is going to last?
Oracle: As long as it can. What about the others?
The Architect: What others?
Oracle: The ones that want out.
The Architect: Obviously they will be freed.
Oracle: I have your word?
The Architect: What do you think I am? Human?
So apparently even though the Architect wasn't optimistic about the new truce working out, he didn't even consider going back on the agreement he had made with the now-dead Neo, even though it presumably would have been within his power to do so.