Short Answer : No.
Long Answer (It could get long, but I will try to keep it as short as possible ) :
1st of all, people in the matrix originally (as in at birth and in general life after birth) do not have a free mind, cause if they did they would be able to do what Neo does by default (Morpheus: Free your mind, Neo. Neo: Free my mind. That's right, free my mind. Okey dokey).
Therefore, yes IF people had a free mind then the "emergence" of The One would be "a totally random event" infact the system would crash under the circumstance coz everyone would "potentially" (see how the word is used in the Movie, and the games) be the one. But only a select few have the ability to see beyond the ball and chain programming of the matrix and break free (as in "Free their mind"). The more you are able to bend the chain (the rules) the more closer you are to the ability of The One, and if you break the chain then you are The One and everything below follows (just like the six previous versions of the Matrix; explained below).
Even The One is just another method of control like The Oracle and the System, because he is accounted for and used like a Trojan Horse unbeknownst even to himself until he meets the Oracle in the third part and she tells him.(Oracle:Because it wasn't time for you to know. Neo:Who decided it wasn't time? Oracle: You know who. [She points at the Temet Nosce sign above the door])
And as a statistical genius that the architect is and in turn the machines and The Oracle are (shown time and again in the movies, how the oracle can predict what might the outcome be) they have gathered that once The One has emerged and "freed" the rest of the anomalies, it would be time for destroying Zion or the system would crash as the number of anomalies increase exponentially after the "Emergence"(Morpheus: We have freed more minds in the last six months the we have in the past six years.). For this purpose they also have taken into account all possible responses that this One anomaly would have to all the different situations, emotions etc. and you can see it in the screens as The Architect analyzes them while providing just the right information that is needed by him to get the right response.
Statistically The One, being human, would have sympathy towards his species and would want its survival and would comply with need of the machines i.e to let Zion be destroyed and the Matrix restarted after choosing 16 females and 7 males.
There is a lot more where that came from, but like I said I am trying to keep it short.
EDIT: TO hell with short...
Now if you have read/watched any of the Wachowski Brothers interviews about the movie you may know that they have borrowed heavily from the Hindu (Old/Original Name: Sanātan).
According to Sanātan Philosophy the entire creation is Nothing but Maya (Illusion), and we humans (our Souls, Atma) are like prisioners in this illusion and once we realize this fact, we take the Path to Self Realization (Know Thyself). And only after complete realization (Mind Freeing) will we actually Merge with the Param Atma (The Source of Creation)
Keeping that in mind, know that in the Hindu Philosophy there are three parts to the cosmos, The Holy Trinity, consisting of The Creator, Brahma; The Maintainer/Caretaker, Vishnu (AKA Hari); and The Destroyer, Siva (AKA Shiv, Shiva, Rudra (Old/Original Name);
Now Brahma is one who created the cosmos. In the beginning he created a world where there was no death. Everyone was happy and well fed, the perfect world. But soon it all turned into chaos as it would if there was no death and people kept reproducing also after a while being well fed and nothing to worry about does get mundane. Therefore he created Vishnu and Rudra; One to maintain and One to Destroy (and thus recreate).
When Rudra, The Destroyer/Recreator was told what duty he would be performing he wild with rage, but was pacified by Vishnu when He explained to Rudra that in essence His role was that of a Recreator as Recreation cannot occur without Destruction.
This is the essence of the relationship that The Architect, The Oracle and Neo (The One) shared; they are the Trinity (see how the name Trinity is used in the movie; that Neo was is love with and that which mattered to him the most).
P.S.: There is even more...
The Bhagvat Gita,