The reason to send Kyle Reese back in time to become John Connor's father is because he is John Connor's father and has to be sent back in time in order for that timeline to exist.
Also, please remember we're talking SciFi here… with time-traveling and a perfect example of a so-called "predestination paradox". For John to exist and send Kyle back, John has to "create" Kyle… creating that so-called predestination paradox.
A predestination paradox (also known as "temporal causality loop") is a theoretical phenomenon, which is said to occur when a chain of cause-effect events is circular; for instance: if event A causes event B, and event B causes event C, and event C causes event A, then these events are said to be in a causality loop.
I might as well ask back: "if you can tell me how to solve the predestination paradox, please post an answer" because no one could explain or prove the Bootstrap paradox yet! The "logic explanation" or even "proof" you're asking for is impossible to provide because whether or not a scenario described in this paradox would actually be possible, even if time travel itself were possible, is not presently known.
As soon as time traveling becomes possible, I'll gladly check on the paradox for you. Until that time, it is a given fact that it is impossible to state where the original originated. Thinking of it, you could compare your question with the quest to the answer which was first... chicken or egg? One thing is for sure: one was first.
Terminator (first movie) says Kyle travels back through time to protect Sarah. Sarah isn't pregnant before they meet. The movie tells and shows us, it's Kyle who was first and created "a new future" by not only protecting Sarah, but also making love to her in the tunnel where they hide out and spend the night. That causes Sarah to become pregnant as shown at the end of movie as she drives off into the desert with a fat belly and records a first message for her yet-to-be-born son… while a storm is coming up.
The movie even keeps ample room for interpretation that John planned his own birth by sending Kyle and not somebody else. But that was never really explained in any of the Terminator movies, so it's just room for interpretation… nothing more, nothing less. If you're willing to accept time traveling in SciFi, you'll have to accept it comes with a "bootstrap paradox", a "reverse grandfather paradox", a "predestination paradox", and some exceptional contradictions to the second law of thermodynamics. That's SciFi for you... it has Fiction in the Science. ;)
Thinking of it, there's another great example of the paradoxes involved in the time traveling we see in Terminator movies.
Remember the part in the first Terminator movie where Kyle tells Sarah that John will hold that "big speech" in the future?
- Kyle (the dad) tells it to Sarah (the mum),
- Sarah tells it to John (the son), and
- in the future it will become a big speech by John directed to all humans, and Kyle will hear that speech for the first time (according to what Kyle tells Sarah while talking about John's speech).
- Kyle travels back in time to tell it to Sarah (see "1.")
Welcome to another causal loop. Who wrote the speech? Common logic implies "no one… it exists all the time without a clear indication where or when it origined".
If you really want to wrap up all theories which might or might not explain the time travel paradoxes in the Terminator movies, you might want throw in total confusion by checking the end of T2 which states "The future is unknown; and the only future is, one we make for ourselves" which could very well indicate that John might also have influenced his own "present" and "future" by sending Kyle back to the past.
And looking at the end of T1… a pregnant Sarah is traveling through Mexico. Along the way she records audio tapes which she intends to pass on to her unborn son, John. She debates whether to tell him that Kyle is his father. A boy takes a photograph of her which she purchases (it is the photograph that John will later give to Kyle) and then she drives on towards approaching storm clouds.
It's pretty much reasonable that she is in doubt if she should tell John about Kyle because it would make the paradox obvious and it could influence John in maybe not sending Kyle back into the past… which would mean John would never be given birth to by Sarah later on… and without John existing (and leading the resistance) who knows if humanity would survive at all.
Looking back, the Terminator movies are full of predestination paradoxes…
In T1, the cyborg T-800, sent back in time to assassinate Sarah Connor, is destroyed… but its parts are salvaged, for the formation of the Skynet network which leads to Skynet becoming self-aware in 29th August 1997. Kyle Reese (who has been sent to protect Sarah Connor), ends up fathering John Connor (his future commander) with her. The former predestination paradox, however, is prevented by the characters' choice to destroy all of the remains of the Cyberdyne Systems, thus breaking the loop.
In T3 however, the Terminator T-850 sent back to protect John from a new powerful T-X killing machine, states that the events of T2 only postponed the date of Judgement Day to 24th July 2004 at approximately 18:18 — but never stopped it, and that Skynet's rise is inevitable. He turns out to be right, as the movie ends with Skynet coming online.
I couldn't help adding a quote of the script of "The Terminator" (first movie) to point to the answer if Kyle Reese always was John Conner's father:
Sarah speaks quietly into a hand microphone as a dark-complected attendant laconically fills her tank. She cradles the cassette recorder in her lap, in the lee of her SWOLLEN BELLY. She looks to be about SIX MONTHS ALONG. Under her down vest she wears a leather shoulder holster and the butt of a .357 REVOLVER presses against her breast. She tugs the vest closed as the attendant glances her way. A German Shepherd sits in the back among taped boxes and suitcases.
Should I tell you about your father? That's a tough one.
Will it change your decision to send him here... knowing?
But if you don't send Kyle, you could never be.
God, you can go crazy thinking about all this...
I suppose I'll tell you... I owe him that.
And maybe it'll be enough if you know that
in the few hours we had together
we loved a lifetime's worth...
Now tell me... who's more trustworthy to give an answer than John Conner's mom? ;)
Oh, and if you want to know how logic makes that possible - read this answer to another Terminator question which provides a nice insight on why it was Skynet who actually *created John Connor by sending the first terminator back to kill Sarah... resulting in the resistance sending Kyle back to protect her and - by doing so - making John's creation and birth possible in the first place.