Why did Protagonist want to know who recruited Neil?

Warning: this answer contains spoilers for Tenet.

Throughout the film, the Protagonist becomes increasingly concerned by how much Neil seems to already know about what's going on. Neil often displays more understanding of the events than the Protagonist himself, even though—from the Protagonist's perspective—Neil shouldn't have been involved with any of this before the Protagonist hired him to get close to the weapons dealer. (The moment when Neil refers to "the cavalry" being on their way makes this especially clear—the Protagonist has no idea who Neil is talking about, and so he realizes that Neil must already be much more deeply involved than he knew.) At a certain point, then, the Protagonist becomes convinced that Neil had already been recruited into this strange world of temporal espionage, and he concludes that someone else had already recruited him (and thus the Protagonist is probably worried that he's been set up). The Protagonist is only half right:

The direct implication is that at some point

In the scene you're quoting, Neil probably says "It can't possibly do you any good to know that right now" because he's walking a fine line here—he's trying to reassure the Protagonist that they're on the same side, but he also knows that "ignorance is their weapon," and that the Protagonist still needs to figure out for himself the truth about his own role in the organization. (That's my interpretation of that comment, anyway.)