There are two fundamental things that are (or should be) at play here: damping and natural pitch stability.
First, stability. Most airplanes (but not all fighters) have a natural tendency to come back to a certain (called trimmed) angle of attack. I won't explain here how it happens aerodynamically, but the point is that as the airplane pitches up, a moment arises that pitches it down (like a spring). This applies to any disturbance, be it a wind gust or an elevator deflection. So pulling (and holding) the elevator will not produce permanent rotation (unlike, say, the isolated roll response to ailerons). Instead, the airplane will settle on a higher angle of attack (AoA). (This will, of course, increase lift and the aircraft will depart from its former trajectory, but this is another matter).
Typically, you model this 'spring' by having a special function $C_m(\alpha)$ (pitch moment coefficient as a function of AoA (or of lift coefficient)). For a stable aircraft, it is linear and negative in the operational range of AoA. Which simply means: more AoA -> proportionally more pitch down moment. This is one of the most important functions in the whole flight simulation.
Elevator deflection (say, up) will add a pitch up moment. This will upset the original balance and will cause the airplane to accelerate pitching up. But this will only keep happening until the moment from the elevator balances out the growing pitch down moment from natural stability.
Now, the dynamics of this pitch process depends on several factors. Apart from this $C_m$ and the elevator deflection, it will depend on damping. Damping also happens naturally, and it happens for all airplanes, whether they are stable in pitch or not.
Damping, by definition, is the process that counteracts velocity: that is, the faster you move, the more the force that slows you down. In the absence of other forces, you will settle on some constant speed after initial acceleration. This, again, is something that would happen in an isolated roll motion: when the flow is symmetric, the wing doesn't produce any spring-like forces that restore bank.
There are two generic ways to simulate damping. One, you just directly introduce this term: $C_m(\dot\theta)$, i.e. pitch moment as a function of pitch rate. In the normal operational range of AoA, this is nearly a constant (i.e. a proportional term), and is always negative. The value of this constant is fairly easy to estimate from the tail geometry - see below. But things get more complicated as you approach stall. (They always do).
The second way is a simulation of how damping actually happens. Most of it comes from the fact that the local AoA on the tail changes as the aircraft pitches. Consider this: when you are pitching up, the tail goes down. More air "comes from below" to it, which is equivalent to higher AoA. The level of this increase depends on the added vertical speed of the tail with respect to the normal forward airspeed: $\arctan(w/v)$. (In practice you can usually drop arctan as the angle is small). Now, $w$ linearly depends on the pitch rate $\dot\theta$ (often denoted $q$) and the "arm" $L$ of the tail (its distance to CG): $q\cdot L$.
So, if you simulate the tail "properly", i.e. simulate lift on it and through that all the moments it produces, you'll need to add this local increase of its AoA, and everything will happen automatically. Alternatively, if you simulate the moments from the tail directly (which is often done), you can derive the amount of damping from knowing its nature: it (or rather its coefficient) will be proportional to $q\cdot L$, then to another $L$ because the force acts at this distance (making the dependency on $L^2$), and to the area of the horizontal stabiliser, and inversely to speed.
Tail is not the only contributor to damping, but usually it's by far the main one. If you account for it, your model will behave much better. Still, to account for all other things, you'll often have some additional term $C_{m_{other}}(q)$ (at least just linearly: $C_{m_{other}}^q\cdot q$), even if you simulate the tail separately.
This hopefully resolves the question why the model behaves so unnaturally. It remains to consider what happens if you indeed want to simulate a statically unstable fighter.
A statically unstable aircraft will still have the same damping, but its 'spring' term $C_m(\alpha)$ will be positive (or zero for neutral stability). If you pull the elevator up, the aircraft will start pitching up indefinitely. This motion will only be restrained in three ways:
- Damping will limit the maximum pitch rate.
- As the aircraft reaches stall (esp. tail stall), things will get very different. The aircraft may become stable again (leading to the Cobra-like manoeuvre), or may get "stuck" at certain high AoA. Simulating these regimes is a separate problem in flight dynamics.
- Active computer control. Yes, a simple PID loop can solve it. Yes, delays in the control system is a real problem, and you will need more careful tuning of the controller, esp. its D term. It's fun.
