Will an airplane engine failure on takeoff cause an aerodynamic stall?

But your sample size is too small.

True.

But I believe even the CAA, predecessor agency to the FAA, had some requirements for longitudinal stability for airplanes. Current FAA requirements state:
https://www.law.cornell.edu/cfr/text/14/23.2145

§ 23.2145 Stability.
(a) Airplanes not certified for aerobatics must -
(1) Have static longitudinal, lateral, and directional stability in normal operations;​
(2) Have dynamic short period and Dutch roll stability in normal operations; and​
(3) Provide stable control force feedback throughout the operating envelope.​
(b) No airplane may exhibit any divergent longitudinal stability characteristic so unstable as to increase the pilot's workload or otherwise endanger the airplane and its occupants.

Also:

§ 23.2150 Stall characteristics, stall warning, and spins.
(b) Single-engine airplanes, not certified for aerobatics, must not have a tendency to inadvertently depart controlled flight.

And while the following Part 25 longitudinal stability requirements apply to transport category aircraft, I believe many Part 23 happen to meet the same stability requirements which are even more stringent:
https://www.law.cornell.edu/cfr/text/14/25.173
 
Great research and write up! However the C-152 is a docile airplane that really doesn’t want to kill you. I’m not surprised at the result, but I find your excuse that the FBO had no 747 to rent a little thin. 😋 We need you to repeat this experiment with many other types of crafts. 😁
Ha!
I actually intended to do the same with a C-172 but after I finished the experiments in the 152 I lost interest. Never got around to publishing my results until now.

I do recall looking to see if anyone else had ever done something similar and posted results to Youtube around that time and did find one person who had. But it hadn't got a lot of views. Don't know if it still there.
 
It seems that this subject is also ripe for a good dynamical analysis. And that a simulator, such as X-Plane, does a fairly good job of simulating actual flight dynamics. Perhaps someone would like to give it a try? This would provide an easy way to test multiple airplanes. And a C-152 simulation could be compared to Jim's results.
 
One note on the airplane's configuration I forgot to mention is that I wanted to push the CG aft so it was closer to the center of lift. My theory is that should slow down the speed at which the nose would drop. Since it was a "scientific" study I picked a piece of scientific gear as dead weight and shoved it to the back. It was a Tektronix 465 Oscilloscope weighing about 25 pounds.
s-l400.jpg
 
I thought I had lost these files, but found them in a directory with the wrong date. I had forgotten that I had also placed a folding bike in the back of the C-152. Even with those weights in the back you can see the CG is still in the middle of the allowable range. With just a student and CFI the CG is fairly far forward in those planes.

(I was curious whether the folding bike would fit in the back of the C-152, hence my choice.)
 

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Side note: the folding bike cost $250 when I bought it in November 2013. The company was fBike and sold a variant of a Chinese made bike. The company itself eventually folded. Attached are some photos of the bike when I tested to see if it would fit in the baggage area of a C-172. This is before I bought the carrying bag that I used in the C-152 engine-out tests.
 

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