first, I don't know exactly whta "sequence of infinitesimal Lorentz boosts" you're referring to. A boost of the flat Minkowski space gives you a Minkowski space back, so you can't get a curved space by any sequence of boosts that act globally on the spacetime.
Also, the adjective "infinitesimal" could be pretty much inconsequential. Transformations in physics are often represented as the product of infinitely many infinitesimal transformations.
On the other hand, the statement that the world - the Solar System - may be approximated by special relativity is obviously true. The curvature of spacetime is small and may be treated perturbatively.
In pretty much any reference system you choose, and you may even choose the geocentric system, all the gravitational effects including the general relativistic effects may be incorporated as small corrections.
In particular, you may always define some operational coordinates in which the spacetime is nearly flat, so the metric is given by the flat Minkowski metric. The actual metric dictated by general relativity, whatever it is, may be expressed as a small correction to the Minkowski metric.
In principle, such a correction may be calculated at arbitrary accuracy - not just the leading level - by perturbative expansions. This approach to general relativity is usually referred to as linearized gravity - even if one goes beyond the linearized level - and it is not only essential for the actual calculations of observable phenomena but also for a proper conceptual understanding of general relativity (and quantum gravity - the linearization is necessary to understand that there are gravitons, and what their mass, spin and physical polarizations are).
Einstein himself calculated the effects of a point mass - e.g. the Sun - on other bodies - such as the Mercury - in the perturbative scheme. It was before Schwarzschild gave an exact solution for Einstein's equations in the spherically symmetric case (when extrapolated everywhere in space, it is now referred to as the Schwarzschild black hole).
The effects that the GPS has to take into account include, of course, the standard classical mechanics forces such as the centripetal force or the Coriolis force, as well as some special relativistic and general relativistic effects. But of course, when things are calculated in practice, people are de facto assuming that the whole world takes place on a Minkowski background and all the gravitational effects are small corrections that occur on the Minkowski arena.
Even if Einstein hadn't discovered GR by methods of an ingenious physicists, engineers working for GPS would later find the right corrections in the process of adjusting their system and removing the discrepancies. They could experimentally measure the dependence on the daytime, latitude, and longitude of any effect that matters and that made their previous GPS plus software inaccurate. They wouldn't appreciate the beauty of GR right away, but of course, they could guess the right form of the correction terms from the experimental deviations.
Again, to answer your last question, the perturbative approach to GR, expanding around an SR background, doesn't have to fail anywhere. It may be exact just like $\exp(x)$ may be exactly expressed by the Taylor expansion, $1+x+x^2/2+x^3/3!+\dots$. Of course, the perturbative approach becomes awkward, to say the least, for causally nontrivial setups such as black holes - including the interior; wormholes and other topologically nontrivial shapes of space; or some very global questions in cosmology. But when the spacetime is topologically trivial, perturbative expansions work. The flatter the space is, the more useful the perturbative expansions become.