Interesting Thoughts about TS

Why would Tony Stark give the E.D.I.T.H. system to Peter Parker and not War Machine, Nick Fury, Captain America or Falcon?

Here's a picture of the Avengers as of Captain America: Civil War (plus Bucky).

Tony was still angry at everyone on the left, especially Cap. He felt that Thanos only won because Cap wasn't there when they needed him. He said so when the two met face to face. And he felt the others had made the wrong choice.

War Machine was under the control of the military. If he had been given E.D.I.T.H. to him, it would have been turned over to the government.

Stark didn't trust Fury.

Vision was dead, the Black Panther was head of a foreign government and the Black Widow had turned on him.

Spider-Man was the only one he felt he could trust.

But, he has to have made that bequest before the final battle with Thanos. He was hoping to survive and continue to mentor Peter.

Was it too much to trust to a teen-ager? Yes, but he'd already given Peter a spider-suit that could kill on command and was prepared to invite Peter into the Avengers. Either he saw that Peter could be trusted despite his youth or he had a major blind-spot about Peter.

— Edit and Spider-Man: No Way Home Spoiler—

Tony probably recognized that Peter is the sort of person who will do what's right regardless of personal cost. In Infinity War Peter insisted on going along into space with Iron Man to rescue Doctor Strange.

In Spider-Man: No way Home Peter proved that his moral center is better than Doctor Strange's. While Strange was willing to send five people back to their own timelines and almost certain doom, Peter insisted that they could be "fixed" first, even if it meant fighting Strange and risking his own life. This justified Stark's confidence in him, even if Tony wasn't there to see it.