Being a band director, a trumpet player and a musician, I agree but with some caveats:
Not everyone gets stronger through the course of "the season". Although some of what you describe is actually "growing stronger" much of the relative ease of the show at the end of the season is learning how to pace yourself through the show.
It is one thing to be challenged but, it is another to be given the tools to help you meet the challenge. Many band directors and private teachers know what they'd like you to do but do not have a clue as to how to get you there. Students with poor embouchures or bad practice routines who are not given a plan for improvement will not improve simply by putting Maynard's "MacArthur Park" in your folder. These students are doomed to be among the legions of 'basket cases' that the "modern" music education system produces. In 20 years, you'll be back here in a forum as a comeback player having not playing in 15 years.
I DO believe that anyone can play high, much as anyone can run or swim as has been described earlier in this thread. Some people are can intellectually work it out and be quite functional. Some people are gifted and have no idea how they do it. It is one of life's little obscenities.
Personally, I have "gotten serious" about playing and in the last two years (in my 50's, mind you, finally gotten past the fourth ledger line G and can daily and repeatedly play up to DHC and beyond). OK, I got religion and started working with a teacher with a "system", DAILY and am making demonstrable improvement.
I think one of the most important things is to find a teacher/mentor/guide who can get you from where you are to where you want to be. Just because they can play well doesn't mean that they can teach well. I didn't improve one iota in the 25 years from the end of graduate school until I "took the plunge". If I do enough soul searching, I can even make a case for saying that I didn't get better - technically - as a trumpet player between 1975 and 2009. Sure, I learned the literature, solos, transposition and orchestral excepts (very important for the school band director) but not so much about range, endurance, flexibility, airflow etc. I learned about much and not so much about how to make music on my instrument. I am now at a point where technically, I feel there are no limits and I am no longer the slave of the trumpet but the trumpet does what I say. I am not THERE yet but I'm on my way.