Granted, there's a lot of snobbery in the area, but I'm not sure that's the whole story - our neighbours did public all the way K-12, they just wouldn't touch HHH. The 2 girls went out of area (neither went to HHH or Riverside). I don't know how / why HHH became so undesirable.
It was undesirable when my Dad went there and still seen that way when I went there back in the 90s.
I have no idea what it's like these days OP but have very fond memories of my time there so thank you for giving me the opportunity for a trip down memory lane.