Tony wanted to recruit Hulk but he wasn't available, Black Widow wondered if Hulk would even be on their side. Tony didn't just recruit a 15 year old, he was looking for someone more powerful then him, with real superpowers. I think the Russo bros sort of assumed the viewers already know Spidey isn't going to die with his healing power, and didn't try to explain why Tony felt comfortable sending him to fight. If Vision blasted Spidey right in the chest he could have just walked it off, can't say the same about his other teammates though. There was not a single moment when Spidey was put in danger. Vision and Spidey were the only two who were completely harm free during that fight.
After the fight Tony called Aunt May to pick Spidey up, he was done. They had to make a point that Spidey is not an Avenger. He was only tagging along because they were outnumbered. That's how Spidey is usually written in other people's stories, he shows up to help a friend then leaves after the work is done. That's the kind of relationship he has with other Superheroes.