It's virtually impossible to not hit your forearm assuming you are doing everything correctly. If you look at all the high level recurve comps, everyone has an an arm gaurd. The same is true for chest protectors.

Google videos of the World Champs or Oylmpics and you'll see what I mean.

Even if this were not the case, why risk it? You don't want to get hit by a bow string on a 44lb bow in the middle of a comp.

When I was shooting bows, if you did things properly, you wouldn't hit your forearm. The problem is if you blow chow and don't notch the arrow properly. Or the notch blows out. Or the arrow shatters when you let loose. Then the string will hit your arm. I got hit once with a 35lb bow. I don't want to get bit by a 60-70 lb bow string nohow no way.

The angle between your arrow and bow arm may have been too high.

The strongest position is when your arms and shoulders form a straight line (with your shoulders low). So the closer to parallel the line of the arrow is to this, the easier it is to hold.

Now, technically you are correct. If your release is perfectly clean there should be no problem. But in reality, there will be some degree of 'plucking' going on. Your fingers can't open fast enough. So the string actually vibrates a bit from side to side. I have seen some weird follow throughs that avoid this problem.

There are very few top recurve archers that shoot without a guard.

arm guards are worn by all Olympic-style archers (there are some traditional archery styles where they are slightly less common), but wearing two arm guards is something you might see at summer camp before the counselor gets the chance to show how the equipment should be used...

Hawkeye uses a bow in super powered battles. Presuming you won't always have the best form under fire just seems sensible.

Also, what if he needs to shoot... "goofy"?

AIUI, you're exactly right. He wore two arm guards because he kept hitting his arm. Of course, my source for this is reddit comment threads back when Avengers first came out, so make of that what you will.

That was my source as well...

In the picture you linked, he's not wearing two arm guards. He has an arm guard on his left arm (the one holding the bow), and a finger guard on the right (the one drawing the string).

I think parent was referring to the two strips on his bow arm (which is his right arm, btw; he is shooting goofy, or the photo is reflected.)