I'll try to approach it from a slightly different angle.
Fundamentally, there is no difference between light aircraft and airliners in terms of stability. 'Normally', that is, on the 'front' of the power curve, if the aircraft is statically stable in pitch (or more precisely, in angle of attack), it will be stable in speed.
If you have a classical trim tab and a free-floating elevator (reversible control), it still behaves largely like an irreversible ('fixed') stabiliser/elevator: as you accelerate, the aerodynamic forces on the elevator and on the trim tab increase proportionally, and the hinge balance remains. (Of course, assuming that the changes are not too great to alter the flow dramatically). There will be a difference in pitch stability with relaxed control, because the elevator will 'flop', but in the end the steady state will be the same.
However, there are substantial 'practical' differences between the light and heavy aircraft which make them behave differently in this regard.
- First, as already mentioned, the thrust line on most modern airliners passes quite far from the centre of drag (and CG). This does matter for the speed changes caused by thrust - which are, unlike pitch changes, are the most practically relevant changes. (It is rare to encounter a wind shear of such duration that speed would re-stabilise). Having thrust line low enough can completely destabilise trim (with respect to thrust changes). In fact, the 737 MAX ordeal is a testament of how important such considerations are.
- Second, and this is more interesting, the so called long period motion (in particular, speed) on heavy aircraft is more decoupled from the short period motion (e.g. pitch) than on light aircraft. In other words, the speed and altitude dynamics is slower with respect to the pitch/roll/yaw dynamics.
This latter deserves some discussion. For the pilot, this makes control more difficult in some ways - but in others, easier. This all depends on the task at hand.
In particular, if we just change trim, a light GA airplane will very quickly find a new equilibrium with speed and climb angle. But a heavy will (naturally) quickly reach the new AoA, and then enter a long sequence of very long period phugoid oscillations. Here engineers can help pilots to make control convenient as
they we want.
Enter human factors. In general, research tells us, we achieve the best results if we control the first derivative of the target parameter. Say, if we want to aim a gun, we want a joystick which controls its pitch and azimuth (or yaw if you like) rates proportionally to the applied force.
So, what control is the 'best' for an airplane? Of course, this depends on the aim. Many modern control systems reconfigure themselves for different tasks. But for 'normal' flying, we mostly want to control the flight path angle. This is one integral away from the load factor, and this is why (to a large extent) Airbus stick controls the load factor.
Great? Almost. Not quite. There is one problem with humans. We naturally predict things. When we interact with the world - we walk, we see - we constantly and subconsciously forecast the world the way it's going to be in a few moments. As a result, when the motion is reasonably slow and does not exceed our abilities, we prefer a seemingly more 'difficult' way of control - directly, without an integral between the input and output. Or a blend of them. And we actually get measurably better results. For our little task of setting speed, setting trim (or elevator control in general) for speed control is more 'direct' and faster than thrust control, even though the underlying dynamics may be more complicated. This may be counter-intuitive for most people, but experienced pilots may prefer that, and this is why (my guess) Boeing made it that way.