YES YOU CAN!!
As I said in the question, I used this recipe http://www.joepastry.com/2008/french_buttercream/ .
Before I embarked on the experiment, I whipped egg yolks with my hand mixer and in my food processor. Since there seemed to be no difference, I proceeded.
The first picture was before whipping, the second was whipped for 5 minutes with the food processor, the third picture was whipped for 5 minutes with a hand mixer. Prior to whipping the yolks measured 1/3 cup. Both whipping methods yielded just over 1/2 cup after transferring the yolks to a measuring cup which broke the biggest bubbles.
While the yolks were whipping, I made my syrup and pulled it off the heat at exactly 148F, and transferred it to a pyrex measuring cup. It was at the next step that I first saw an actual advantage to the use of the food processor vs a stand mixer. There was no need to stop the machine, pour, stop again, pour... I just poured in a slow steady stream until all the syrup was incorporated. One thing I noticed after that is that it was taking forever for the mixture to cool, so I wrapped the bowl in a cold damp towel.
WHOOPS! This isn't supposed to happen!
See the hole at 2 o'clock? That's a finger press mark in the still not quite cool mixture. At this point I figured I was toast, but I'd come this far so I kept going. I turned off the machine, took off the lid, and just waited for it to completely cool. When the mixture was completely cool (almost an hour later), I restarted the machine and started dropping in the softened butter. After about a stick I used a butter knife to scrape off the hardened mixture that was still adhering to the blade and bottom of the bowl, and just kept going. When the hardened bits finally dissolved (at about halfway through the butter), I switched from the metal blade to the plastic blade of my food processor. I should note at this point that I found the same advantage to the food processor here as when I incorporated the syrup, the feed tube and lid made it really easy, I only stopped the machine once to scrape down the sides.
OOH! It's starting to look like buttercream!
The peaks in the bowl were created sicking in the whisk and pulling it out. Because the camera was giving me a hard time, the pictures of peaks were actually taken 10 minutes after the peaks were created.
At this point the frosting was delicious and of a perfect constancy to frost cupcakes by the dip and twist method. Since what I really wanted to do with it is use it as a filling, I took SAJ14SAJ's advice and gave it a 3 minute whip with my hand mixer. That did give it a slightly more matte finish and a bit more structure.
The proof is in the pudding. EEK is that a BOXED cake?? hangs head in shame I refrigerated the bottom layer with the filling (which of course significantly solidified the filling) before placing the top layer, then I let the whole thing come to room temperature, frosted the cake, refrigerated, and smoothed the frosting with a hot spatula.
The chocolate frosting in the final picture was made with the experimental buttercream mixed with melted bittersweet chocolate and powdered sugar.
I don't think that I'm going to have replaced my stand mixer in time for the next time I want buttercream, but my results with this experiment are such that I think I will do it again. The one thing I will do differently next time is that I will stop processing every 3 minutes during the cooling stage, remove the lid, and refrigerate for 2 minutes, repeating until the mixture is cool. That was the only point at which I became less than optimistic. If I do it again, I will update with the results of the procedural change.