There is a general rule: don't trust "Pilot's Handbooks" when they talk about aerodynamics. More often than not, they are full of misconceptions and, at best, oversimplifications. In your quote, for example, the conclusion has nothing to do with the preceding text.
But the question is fair. Thrust line can affect stability, but we need to define what exactly we understand by that.
Stability as such is a negative reaction to a disturbance: a disturbance of a particular defined kind must cause a reaction which negates the original disturbance. Usually, we restrict the analysis to relatively small disturbances, when reactions are more or less linear.
Longitudinal stability, in a broad sense, describes any longitudinal motion. For airplanes this involves pitch or speed. (Altitude is a different axis and even a different frame, although we'll need to consider it in some cases). These are linked but different motions, and we can talk about them separately to some extent. It just happens that for most airplanes, the characteristic times of these angular vs linear motion is so different that we can very well analyse them independently. The heavier the airplane (and/or higher its wing loading), the better it holds. Even for GA airplanes it's still largely true.
In a strict sense, when aircraft designers (but not pilots) talk about "longitudinal stability", they mean the short-period pitch stability, or more accurately, angle of attack (AoA) stability. This means that when AoA is disturbed (by a gust or control input), a moment immediately arises that counteracts this AoA change (which is done via pitch change).
The mechanism behind AoA stability involves purely aerodynamic moments/forces. (The explanation involves the concepts of neutral point and moment derivatives). Notably, it does not involve thrust nor airspeed. Both of them change too slowly compared to AoA/pitch and so play practically no role in longitudinal stability per se.
But when it comes to airspeed stability, with which pilots are more intuitively familiar, the situation is different. Airspeed stability is linked to AoA stability via an indirect mechanism that Robert already roughly described in his answer: "increased thrust, increased speed, increased lift, plane rises vertically, vertical "up" pushes tail down, pitch changes". The important result is that a statically stable (read: AoA stable) aircraft will also be stable in speed. But even that holds only if drag doesn't grow faster than lift; that is, on the front side of the power curve. (Pilots know it very well). At high (but pre-stall) AoAs, the airplane will remain stable in AoA but will become unstable by speed.
Technically, this is still speed stability, not thrust stability. We are not concerned how airspeed changed: thrust, gust, dive, whatever. A statically stable airplane will try to pitch up and climb as a response to increased airspeed, slowing down as a result.
But when we analyse speed changes as a specific result of thrust changes, other factors come into play. Namely, apart from speed, changing thrust can disturb the moment balance of the airplane. In general, multiple effects can be important here, not only the location of line of thrust with respect to CG, but also with respect to 'centre of drag'; and the changed slipstream may cause aerodynamic changes. Either way, it may happen that these extra moments may augment or negate the natural tendency that comes from AoA stability.
For simplicity, let's analyse a few obvious cases of reaction to increased thrust (leaving everything else unchanged, particularly trim).
Thrust is in line with CG. The airplane will start to climb (or reduce descent) and will settle on the same AoA and roughly the same speed.
Thrust line below CG (the case of most airliners with underwing engines). This will produce an added pitch-up moment, which will cause the airplane to slow down more than necessary, despite the added thrust! This is an unstable condition. It can be particularly nasty in go-around situations. How do we fly it? Luckily, unlike AoA, the changes are slow enough for pilots (let alone autopilots) to react with active trim changes.
Thrust line above CG. This is the opposite of the above and, in moderate amounts, can have stabilising effect and easier trim changes. The airplane will settle at a higher speed, which is presumably what the pilot wants (despite the "power controls altitude" mantra). When the line is too high, the airplane may even descend and accelerate more than necessary, until the aerodynamic moment balances out the thrust moment, but overall the condition is stable. (Note that in most cases thrust starts to fall with increased speed, which helps to find the balance).
As a conclusion:
In a strict sense, line of thrust (and thrust as such) does not affect longitudinal stability per se; that is, the AoA stability.
However, it does affect airspeed stability, which many pilots understand as longitudinal stability.
It certainly affects trim changes, which, again, pilots perceive as a measure of longitudinal stability (not entirely without reason).