Finch does not yet fully trust Reese for several very good reasons. Chief among them is this: Reese is not the first operative whose assistance Finch has sought in the matter of the numbers.
A few of the initial names on his list had simply proven unapproachable, despite their excellent qualifications. Frank Moses, for example, has found love since his departure from the C.I.A., reducing the likelihood that he could be as driven a partner as Finch requires. The man currently calling himself Eliot Spencer likewise reapportioned his allegiance in the murky years after disappearing from government employ; a pity, as his redacted records indicate a true gift for the work. Others, however, have proven far more fickle than Finch anticipated given the apparent congruence of their goals.
His acquaintance with Reese is still in its infancy. How can he yet judge whether the man will begin using the means and resources Finch provides to amass wealth and power in pursuit of unrelated personal objectives? Weapons, money, drugs, powerful people with secrets: the crimes they disrupt in their investigations often abound in assets and leverage enough to tempt an insufficiently focused operative off course, as Finch has found to his regret. Reese has also managed to track him down once already; while the incident confirms Finch's assessment of his skills, it also reminds him unpleasantly of another past partner who'd followed him home to ... renegotiate ... their arrangement. Fortunately, Finch has long since arranged the means to protect himself on his own turf.
He likes to think he doesn't make the same mistakes in judgment twice, but he can't afford to risk it. While it might be true that he'll have to trust someone, sometime ... it is equally true that such trust has to be earned.
Reese had not even been the top prospect on the employment list the day Finch had contacted him. While his loss made Reese ripe for the mission, his history of shifting loyalty-– between his nation, his woman, and back again-– had been a cautionary data point Finch could not ignore. Reese's suicidal stunt had made him a more time-sensitive acquisition than Finch's primary choice, however; the drunken antics with fellow passengers had been one thing, but passively allowing the police to detain him afterward had been the equivalent of sticking his neck in a noose. Finch had chosen to step into that closing door, taking the chance that desperation would make Reese more suited to his sales pitch.
So this is how the situation lies: he will continue to provide Reese with only that information required to get the job done; Reese will continue to pry for more while investigating each person of interest; and hopefully, along the way they will better the lives of countless others. Provided they survive long enough, and Reese does not betray him.
Perhaps, one day, there will be room between them for more.
But perhaps not. Finch is, as he has said before, a very private person.