For the same reason you don't send in a SWAT team to handle a traffic violation, its expensive, disruptive overkill and people don't like it (and the US cities who do that are learning the hard way). And the game is rigged anyway.
Expensive: The Matrix is software, and in theory they could inject as many Agents as they like, but we don't know how expensive it is to run an Agent. The Machines have a fixed power budget and will want to be as efficient as possible with it. (Can I hand-wave away the lavish and pointless life-styles of other programs as sequel drek? No? Ok... they're all hiding as parasitic programs...)
Disruptive: The sole purpose of the Matrix is to fool humanity into believing it is reality for the Machines to produce power. Agents go to some length to allow plausible deniability, even fitting ideas of Human "Men In Black" conspiracy theories. Every time an Agent takes over a body or performs a superhuman feat it risks disrupting the lie, it risks harming the "crop".
Overkill: Until Neo, nobody has ever defeated an Agent, and even Neo can barely handle one until his transformation at the end of The Matrix. One Agent will do if all that's necessary is to fight. Agents also search, coordinate with normal people, and chase, so just one Agent won't do.
The Game Is Rigged, Goldilocks The Machines don't want to kill The One, they need The One. To find The One they need Zion. The Machines must allow some humans to escape. They can't make it too easy, or too many people will escape and the people of Zion will be suspicious. They can't make it too hard, or not enough people will escape to form Zion and find The One. Three Agents at a time is just enough.
The Machines would have to strike a balance between Agent effectiveness, cost of running Agents, risk of harming the crop, allowing a trickle of people to escape, and risk of killing The One. A three person team is a very flexible unit and a likely compromise. Once The One appeared for sure, The Machines responded with Upgraded Agents with more license to bend the rules and deal with red-pills, but remained no match for The One (a sort of trial by combat). Even after Neo refused The Architect's offer, they still were not seriously trying to kill him. Maybe they were hoping he would change his mind. Maybe they were hoping to blackmail him with a defeated Zion held hostage. Maybe they were just slow to adapt (The One refusing the Architect's offer appears to have never happened before).
(The parts of this which I consider canon are that Agents are disruptive, that three is generally more than sufficient to deal with red-pills, and that The Machines are not actually trying to kill The One for most of the trilogy. The rest is speculation.)