They made him feel like Robin because as you said, Iron Man was the clear alpha of the MCU when Spidey was introduced. We had spent 8 years with RDJ as IM before Spider-Man came in. You don't just get to come in and be at the top just because. In this universe, being "Spider-Man" meant very little to Tony or any of the other Avengers. It meant slightly more than being "Ant-Man".
I'd like to point out though that for the first time, we're seeing a trilogy where Peter stays in high school the whole time. Yeah, Holland's Spidey has seen some stuff, but he's surrounded by much more powerful superheros in the MCU in Captain Marvel, Scarlett Witch, Doctor Strange, etc (not even counting those that have died). Spidey's skill set as a superhero is much closer to Daredevil than some of those who wield such great power around him.
I hate to break it to you, but Spider-Man is probably best served as a side character. We've just never seen him have a mentor too much because most of the time he's fighting local Queens crime (very small scale crap in comparison to what the Avengers are doing) and he's had to learn on the fly in the movies we've seen him in before the Holland version.