How i would have handled this.
You want a soft reboot, for a new generation. You don't want to ruin what's come before - like the first two.
So, you have JC take out Sky net, in in this movie, then he realises he still has a prophecy to full fill - to send Kyle Reece Back, plus another Terminator for T2.
All he knows, was that a terminator was sent back to 84 along with Kyle, then 2 again to the events of T2, then 2 again for T3. He knows he has to make this happen.
So whilst the troops are happy that sky net is down, he knows his job is not yet finished (that's the problem with time travel). He may wrestle with himself and think - skynet is down, job done, therefor it shouldn't matter - but, obviously, he sticks to his mission.
This is where old Terminator comes into play. He literally comes out of hiding, and helps JC with the time machine (i forget it's name in the franchise) explaining what needs to be done next. He explains that JC should already know that a bad one has just been sent to T1, T2 and T3 periods and they need to counter act it, but also, sky net sent two more that need to be dealt with, as part of the bigger picture.
It's old Arnie that then helps KR get ready to go back for his mission in T1, then he reprogrammes a couple of Terminators for the events T2, T3. This could be a very cool moment with CGI interpretations.
We could see each time travelling and a small montage of those events.
Then, he explains another mission - out in Dubai or say perhaps Asia or Europe and get's JC to send two more Terminators, of any model to certain time zones.
JC doesn't have any recollection of this, but why would he - it was never in his life time, or personal experience.
We then focus on the two Terminators - old arnie is telling them, or perhaps giving him data information of his mission.
We then see from the this new terminator's perspective him going back in time and from there, we have a whole new movie.
Then the sequel could focus on the other terminator sent back.
That way, the time line is intact. No need to re do, or change the past - it's more of a side step, to say, whilst these movies happened, this also happened.
We could see two new, different adventures, where the person being protected could bring a whole different dimension to the franchise.
The Terminator could be any actor/actress.
We can then have a movie of the future, post sky net, see them rebuild a new world.
And the old Terminator - he could be the one that is sent back to dubai or europe and part of his mission is to wait until sky net is down, to help JC. Circle of life and all that.
Personally, the new terminator should be Michael B Jordan
(jokes)