The reason people misspell your name isn't because they can't read, and it isn't because they're unprofessional. The fact is, you have an odd name, and it's unexpected.
People get used to patterns:
I was on my motorcycle commuting to work the other day, and the city just got finished repairing and reopening a bridge that had been closed for a year and a half. The 3 way stop that they temporarily setup was back to a main thoroughfare with a single stop sign at a road that terminates at a "T" at the main road. A lady in a big blue SUV almost hit me because she expected me to stop when making my left. I had right of way, but a week ago I would have had to stop. It's not that she was dumb, although I was a bit pissed off, and it's not that she can't read. I almost made the same mistake a few times when I thought a car was going to stop but instead went right by. For a year and a half, anyone and everyone who regularly commuted through this area had to stop, and just like that, the city removed the stop signs without warning, leading to mass confusion by people who allegedly know the roads.
Patterns in words:
Edit: My real name is on my profile: Adam Arold. They usually spell Arnold. Even if I look at my name like a complete stranger I can see the difference.
Huh? How can you possibly look at your name like a complete stranger? Did you just recently change your name a month ago to Adam Arold, or could your fasinating, mind-boggling ability to recognize the spelling of your name stem from the fact that it's been your name for the last 26 freaking years! ;)
The fact is, people get used to patterns, and when 99.999% of the people whose names look like Arold are actually Arnold, well you can't expect people to scrutinize something this close each and every time. We'd never get anything done because we'd be constantly looking to make sure Webster didn't decide that today is the day we start spelling "the" as "teh".
Aside from the "body memory" involved in doing any activity, consider that there's also a bit of psychology involved when reading words:
Source: Learn English - Can you read this?:
I cnduo't bvleiee taht I culod aulaclty uesdtannrd waht I was rdnaieg. Unisg the icndeblire pweor of the hmuan mnid, aocdcrnig to rseecrah at Cmabrigde Uinervtisy, it dseno't mttaer in waht oderr the lterets in a wrod are, the olny irpoamtnt tihng is taht the frsit and lsat ltteer be in the rhgit pclae. The rset can be a taotl mses and you can sitll raed it whoutit a pboerlm. Tihs is bucseae the huamn mnid deos not raed ervey ltteer by istlef, but the wrod as a wlohe. Aaznmig, huh? Yaeh and I awlyas tghhuot slelinpg was ipmorantt! See if yuor fdreins can raed tihs too.
the fact is that Arold and Arnold look so close that the human mind simply interprets it based on past knowledge of what it's seen before.
In short, it's a bit arrogant of you to think that someone is unprofessional just because you happen to have an extremely rare name that most people don't expect to see.
Lastly, I'd urge caution about sending your proposed letter to potential employers. I doubt that making them feel stupid will prove your worth to them as a future employee; instead, consider something like this:
Hello there! I just wanted to let you know I have an odd name. It's actually "Arold", not "Arnold". It's a common misspelling. By the way, I see you have an opening up for a X. Check my profile out, I think you'll like it!"
In conclusion, use this to your advantage!