Does fly-by-wire permit smaller tail fins (area and thickness) for jetliners? To prevent an XY problem, here's the context:

A striking feature of the 777 is how small and thin its tail fin is compared to the 747.

When I asked about this many, many years ago, someone told me it's because of the 777's fly-by-wire, which makes better use of small fins. But I started to question that recently.



Above are overlaid scaled drawings (747 and 777). Right away the engine placement seemed to provide the answer: one engine inoperative (OEI) controllability.

Using each plane's lateral engine placement and max thrust (282 kN and 513 kN for the 747-400 and 777, respectively), and with a worst case scenario OEI for the 747 (which would be an outboard engine), the 747 ends up with ~27% more yawing torque. Which would explain the tail fin areas of ~77 and 53 $m^2$.

But (big but), I recalled a famous A340-600 OEI takeoff video (a quad just like the 747 and with equal thrust of ~280 kN), and decided to compare it with the 777-300.

And here the tail fin areas are ~53 (777) and 48 (A346) $m^2$:



So, back to square one.

Of course there's the upper-deck (hump) of the 747; if that is one of the main reasons for the bigger tail, then it still does not explain how the A340 (-500 and -600) manages its OEI by having a smaller tail and more outboard engines compared to the 777.

Is it really fly-by-wire that permitted the smaller fins (area and thickness) of the A345/6 and 777?

One possibility that came to mind is perhaps the use of asymmetric spoilers when an extreme yaw is commanded – which can be seen in the linked A340 video at one moment (before/after the ailerons showed big deflection without the use of spoilers). Of course that presents its own problem: the accompanying roll (and with the video's low-quality and lack of telemetry, it is not really evidence).






Tail fin areas via https://booksite.elsevier.com/9780340741528/appendices/default.htm
Scaled drawings via Boeing and Airbus (for the 777 I used the -300, not the -300ER, in case you're wondering about the wingspan).
After much google-fu, for the 777 it's an astounding yes (that someone was right after all).
From a 1996 ICAS paper (see PDF pages 2, 4, and 5) about the 777's innovations, the use of relaxed stability and FBW laws indeed helped reduce the tail area and its weight:



The paper mentions a wheel/rudder crosstie. The wheel here being the yoke. Before the 777 Boeing sized the tail based on controlling an in-flight engine-out using only the yoke. This digital crosstie allowed the smaller fin:

Vertical Stabilizer. Conventional sizing of the vertical stabilizer is based primarily on meeting requirements associated with static directional stability. Sideslip characteristics and the pilot’s ability to control the effects of in-air engine failure using wheel only have sized the vertical stabilizers on previous Boeing airplanes. With the wheel/rudder crosstie, the same desired sideslip and engine-out control characteristics can be achieved with the smaller vertical stabilizer. The rudder input due to the crosstie compensates for the reduced area.


The even smaller fin of the quad-A345/6 remains open.
It turns out my guess had merit. The A340 does use the roll surfaces to supplement the yaw:

The flexibility of fly-by-wire was used to optimize the VMCG speeds. In fact, the rudder efficiency was increased on the ground by fully and asymmetrically deploying the inner and outer ailerons on the side of the pedal action as a function of the rudder travel: the inner aileron is commanded downwards and the outer aileron complemented by one spoiler is commanded upwards.$^1$

And watching the engine-out test video again, this becomes clearly visible:



Note: either the author mixed inner with outer, or Airbus updated it since or flipped them for the -5/600, either way, it doesn't change the answer.

$^1$: C. FAVRE (1994) Fly-by-wire for commercial aircraft: the Airbus experience, International Journal of Control, 59:1, 139-157, https://doi.org/10.1080/00207179408923072 p. 153