While we don’t really know exactly how long Peter Parker has been out there fighting in his homemade suit, the implication is that he’s still fairly new on the scene.
Even in the online videos that Tony Stark showed to Peter Parker in Captain America: Civil War we only saw him just above street level.
It’s not necessarily a phobia, but most of wouldn’t be comfortable enough to just start climbing to the top of skyscrapers.
As Eric Carroll point out, most people have at least a basic fear of heights.