This is arguably the #1 reason why Spider-Man as an Avenger feels off. Even when Iron Man is in LA and Cap is in Washington DC, they never "feel" that far apart. Superman and Batman feel much further apart even though they live within a shorter distance from one another and are even closer friends than Steve & Tony.
The Avengers are weirdly a half-measure between an ensemble team like the Justice League, and a closely-knit team like the FF and the X-Men.
At the end of the day I wouldn't want Spider-Man to leave Marvel, but I do think he would have some advantages as a DC character:One of the cool things I love about the original Superman V. The Amazing Spider-Man crossover is that Superman treats Spider-Man with some amount of personal respect. Now of course he does this because he's Superman and a genuinely nice person but it was still a great gesture in that time.
Batman of course would come up with some kind of contingency plans to take out Spider-Man when he first meets him while Superman will be his usual affable self.
1. He would have New York (mostly) to himself.
2. "First independent teen superhero" actually means something in a world where teen sidekicks are the norm. It doesn't mean as much within the MU.
3. Once you get to Miles Morales, you can do a compare-and-contrast between Peter's mentoring and the militaristic mentoring we see from a lot of DC's characters. Peter is someone who would have a non-authoritarian approach to training Miles, but the MU doesn't leave as much room for exploring that as the DCU does.
4. As I just mentioned, Spider-Man would work better in the JL than on the Avengers.