The basic thing you're missing is the difference between general perturbations and special perturbations.
Special perturbations means
all numerical methods for deriving the disturbed orbit by direct
integration of either the rectangular coordinates or a set of
osculating orbital elements
General perturbations means
methods of generalizing the expressions for simple two-body motion of
a planet about the sun to include the disturbing effects of the other
planets by utilizing infinite trigonometric series expansions and
term-by-term integration
Definitions quoted from Richard Battin's An Introduction to the Mathematics and Methods of Astrodynamics (1999), chapter 9.
Special pertubations methods are called special because when you give them some set of initial conditions, and then you integrate from there to find the state at other times, the answer you get applies only in that one special case.
General perturbations methods are called general because they are solved analytically for all possible initial conditions, and all the integration is done before you ever give it any numbers. What comes out is a set of equations for the state where you don't "propagate" anything: you just look up the answer at the desired point.
Consider, for example, the simple, planar, polar form of the Kepler orbit:
$$r = \frac{a(1-e^2)}{1+e \cos(\theta-\omega)}$$
To use this, you plug in semi-major axis ($a$), eccentricity ($e$), argument of perigee ($\omega$), and true anomaly ($\theta$). Then you read off radial distance ($r$). That's all. There is no need to integrate the equations of motion, because all the math has already been done for you.
This is what SGP4 --- the (U.S. Air Force) Simplified General Perturbations Theory, version 4 --- really does. It is not "propagated linearly with no timesteps", it is implemented as a sequence of nested polynomials and trig functions into which one simply plugs the end time, and the final result is calculated from it in one step.
To make this work, of course, that one step must be outlandishly, overwhelmingly complicated. It is! As David Hammen wrote,
The mathematics of those two line elements is beyond messy. It's a
"math-out". (Think of a blizzard where all you see is whiteness.
Blizzards are white-out conditions. The paper describing the two line
elements is a math-out. All you see is mathematics.)
In that answer, he links a deceptively short article which gives the simplest version I've ever seen. It's only nine pages long, but, well, page five is this:
Now, let me stress again, this is just a small part of the simplified version! To see the original solution in all its glory (or horror, depending on your taste in mathematics), and the nature of the various simplifications that were made in order to make it run fast enough to manage the entire catalog on old computers, you have to go all the way back to Space-Track Report Number Two, "General Perturbations Theories Derived from the 1965 Lane Drag Theory" (1979). That little beauty is sixty-four pages, one of which is
