Honestly, at this point, Peter as a 'young' up-and-coming hero doesn't work
at all. Peter's been Spider-Man for around 12-15 years (realistically should be 15 or more). In-universe, he's been a hero almost as long as the Fantastic Four and technically
longer than the likes of Iron Man or Daredevil or the vast majority of the X-men, along with tons of other characters.
Good point about the cartoons and the Raimi films. The shift seems to have happened somewhere around the 2000's (ironically, in part because of the exponential growth in the character's popularity after the movies). I guess the success of Ultimate Spider-Man might also have been a factor? Suddenly almost every adaptation was about Peter in high-school - including the TASM reboot (as much as I loved that film).
With the films, it's funny (or maybe not) that
every time they try to move Peter's story forward, a reboot hits. Raimi's Spider-Man 3 ended with Peter and MJ as a fully-committed couple almost certainly heading for marriage...then the franchise is rebooted to put Peter back in high school with Gwen as his high school girlfriend! TASM2 ends with Gwen's death and Peter likely heading to college...then the franchise is rebooted to make him 15(?) again and a kid protege to Iron Man! Now, NWH has ended with Peter heading to college and starting life over again as an independent young adult...thankfully it seems
this time, because of the MCU,
that won't get rebooted away in the near-future and we'll be sticking with this Spider-Man for awhile. Or, who knows? Kang Dynasty is coming up after all...maybe the MCU gets rebooted and we're back to Peter in high school