There's no reasonable correlation between spelling and higher intelligence. It can be a proxy for minimum intelligence, absent other information, but that's about all.
Education has many goals and one of them is to teach them spelling. The fact that fourth grade homework is spelled correctly does not matter at all - the goal is not to produce correctly spelled homework.
It won't be embarrassing if no one else in the room can spell better than you. (I once had a Chinese professor forget how to write a certain character while writing on the blackboard. He just laughed it off.)
It's also not clear to me that spellcheck will lead to people forgetting how to spell, since it's all about notifying you of spelling mistakes.
So we should study things so we don't get embarrassed? Moving on are you embarrassed because you don't know how to swordfight or any other outdated skills?
I think in the entirety of my 16 years as a student, I probably wrote less than 100 words total on blackboards or whiteboards. There weren't computers or ipads in those classrooms either.
Whiteboards are something I use more often at work these days, but not really when I was in school.