Now with dialogue!
Just a note, I haven't watched season 8 so if anyone doesn't act in character, that's why.
I really wanted to post this a long time ago but this took a lot longer than I thought~ Basically, I was rewatching Recess: School's out and really liked the last scene with Principal Prickly and TJ. It made me want to make a mirror of it with Twilight and Chancellor Neighsayer cause I thought it'd fit.
While I'm not completely sure on his motives, his desire to keep ponies and nonponies apart is most likely due to how relations between the races had been before hand. Changelings were parasitic creatures that fed off the love of sentient creatures. Griffins were greedy and selfish to everyone, even each other. Dragons were violent and often dangerous. Yaks never made the effort to meet others halfway.
All in all, it's no wonder he had a bad impression of them. Cause lets be honest, they were enemies (save for the yaks) before this point. Dragons are dangerous and have shown that they would harm others for the heck of it. Changelings had only recently tried to take over the nations capital. Griffins… I'm not sure but who knows what relations were like between them.
Things only changed recently (like within a few months) and Neighsay, being old as he is, probably can't keep up with the new shift is opinion. He still lives in the days where changelings could steal your loved one and a dragons mere presence could cause major catastrophe. To live in fear for that long, it wouldn't be easy to change with the times.