Some comments on your extracts from the article and wikipedia:

>> the two theories [quantum theory and classical theory] are completely incompatible

Badly put. For starters, if the background spacetime has no curvature then the two theories are exactly compatible because flat spacetime has Lorentz symmetry at every point, and Lorentz symmetry is built in to the Standard Model of Particle Physics, which is a species of Quantum Field Theory. Secondly, in the absence of an extreme and obviously unphysical universe, there will be a large number of points who have a local neighbourhood which is flat spacetime; this is local Lorentz symmetry, and it is a result of General Relativity (https://en.wikipedia.org/wiki/Lorentz_covariance elucidates, but there's a simple reiteration just before the table of comments there). The Standard Model thus is exactly compatible at least in practically every small patch of any remotely plausible spacetime described by General Relativity. (There exists an enormous amount of observational and experimental evidence for this, and the https://en.wikipedia.org/wiki/Lyman-alpha_forest imprinted on quasar/active-galactic-nucleus spectra is very convincing.)

When one does a gauge fixing [in the https://en.wikipedia.org/wiki/Coordinate_conditions sense], by settling on a way of describing lengths and durations, then one can run into a problem where (a) quantum objects' wavelengths are long compared to the local curvature radius or (b) a very high energy is localized into a very small spatial volume. The (b) case is probably the more vexing: General Relativity predicts the formation of an event horizon; adding pretty much any quantum field theory converts the event horizon into a temporary trapping surface -- also, quantum field theories develop unitarily, whereas unitarity appears to be lost when there is this type of temporary trapping surface (if you make a black hole by smashing together a bunch of atoms, you might in principle get only photons out). The (a) case is really a question of assuming that a locally-useful coordinate charge is globally useful. A flat map of a town can accurately describe angles and sizes of objects on the ground. If you add in some identical very tall rectangular skyscrapers, the (laser-reflectometry/theodolite/etc. measured) distances between the midpoints of any pair's roofs will be greater than the distances between the midpoints of their ground floors. Likewise, flat maps of the whole planet will always show significant distortions of many angles or distances or both. The guaranteed neighbourhood mentioned in the previous paragraph is small compared to the global spacetime, and one has to be aware that ignoring that can distort values for things like scattering angles.

Where (a) and (b) are not in play, the Standard Model of Particle Physics and General Relativity play very well together. In fact, they play well together even when curvature starts becoming pretty substantial, like in neutron stars, where descriptions of arrangements of matter can get weird, but not so weird the Standard Model is inapplicable at any point in the neutron star. Even deep inside collapsed neutron stars (where a temporary trapping surface arises) one can use renormalization: we have tools like https://en.wikipedia.org/wiki/Semiclassical_gravity and various species of https://en.wikipedia.org/wiki/Effective_field_theory#Effecti...

These break down at extremes for a variety of reasons; and the problem is that we do not know if the Standard Model changes at arbitrarily high energies, but we do know that we do not "like" the answer General Relativity gives for arbitrarily high energy in the case where we have a temporary trapping surface, because it makes the far future of the quantum field incompletely predictable.

>> quantum gravitational effects only appear at length scales near the Planck scale

Oops. If you bring a large mass into a superposition of position and then use an accelerometer or put one position inside a cavendish device, we have no good prediction about what the measurement devices will report! The broad view is that they should point to where the mass is, but can't agree on what fraction of the mass is where. There are experiments in play to try to probe exactly this problem of low-energy quantum gravity. Again we run into problems with very large masses: large masses generate an exterior metric very similar to that of Schwarzschild's, but the principle of superposition does not work for such spacetimes (intuitively: two isolated planets, unrotating and starting still with respect to each other, will ultimately move towards each other, following the Raychadhuri focusing theorem; the spacetime between the planets looks less and less like Schwarzschild over time). But quantum mechanically we should expect to be able (in principle) to put planet-scale masses into a superposition of position. If we "measure" the planet (the measurement problem in quantum mechanics is an unsolved pain) do we convert the not-like-Schwarzschild "instantly" into Schwarzschild? We don't know, and the theories hint that they give different answers to that question.

So the problem is "only" in the second bit you quoted. There are various situations in which quantum "smearing" should couple with the dynamical spacetime in which it is observed, and we certainly generate some of those situations in laboratories. But there is a lot of gravitational noise in our laboratories, and we need much larger sets of quantum numbers to have much hope of extracting actual signal.

Perhaps someone will be able to fix that part of the C-class wikipedia article.