Concept of transistor oscillator
The best way to view transistor oscillators is in the frequency domain, rather than trying to apply intuition and imprecise transitions in the time domain. It might make sense for really simple circuits or mechanical systems, but in a typical transistor oscillator there are enough dynamic elements (capacitances and inductances) that ad-hoc hand analysis like that won't necessarily get you far. In fact the journey/derivation I'll present below doesn't consider transistors being in forward active or saturation, and it doesn't even consider transistors at all.
Ultimately, you can view the circuit in front of you as a feedback system. Feedback systems are common and well-understood in electrical engineering -- examples include something as simple as an op-amp amplifier. Generally, we like them to be stable, but oscillators are an example of a feedback network which is intentionally unstable.
In order to discuss how an oscillator can sustain an oscillation, we should start by looking at a stable feedback network first. Here's the stereotypical closed-loop feedback structure, along with an example realization of one. The colors are used to mark corresponding portions of the feedback structure and the example realization I show here.

Let's look at the so-called &quot;loop gain&quot; of the structure. To do this, we null the input, and we break the loop at any point. We then inject a signal, let it make a trip around the loop, and see what we get after it goes around once1:

In an ideal world, the loop gain is negative. However, the loop gain is actually dependent on frequency. Under a typical assumption that the circuit is linear and time-invariant, we can consider its response to every frequency of sine wave independently. For each sine wave you put in, you get a sine wave out at the same frequency. This assumption holds really well for small signals, while for larger signals the amplifier may saturate (not necessarily by saturating any individual transistor).
The loop gain is then a function of frequency, and is composed of both a magnitude (how big the output sine wave is, compared to the size of the input sine wave) and phase (how much the output wave is shifted) compared to the input wave. The loop gain tells us all about how our system responds to a sine wave going around the loop, including whether it grows or shrinks. In the definition of loop gain, I consider subtraction a part of it (since it's both in line with the tools I use, and it is conceptually simpler to think about since there's no &quot;double negative&quot; to keep track of at oscillation).
A sine wave that grows (magnitude &gt; 1) while keeping its exact phase (loopgain phase is zero degrees) is going to become an oscillation, and if there's any such frequency at which the amplifier has a loop gain of zero phase, gain &gt; 1, the system will oscillate at that frequency.
Going back to our ideal world where we want a stable amplifier that doesn't oscillate, we want it to have a loop gain whose phase is 180 degrees -- when we put a sine wave deviation into the loop, the amplifier corrects it.
We can plot the gain and the phase using a so-called Bode Plot:

Notice the orange marker and how it indicates an important characteristic of the amplifier circuit: By the point the phase reaches zero degrees (meaning that you put a sine in and it comes out at the exact same phase), the gain is below 0 dB (meaning that the oscillation dies out). This amplifier is stable.
Now consider the same calculation of your Colpitts Oscillator -- it has a structure that is inherently unstable, and its phase reaches zero degrees (meaning that a sine in creates a sine out at the same phase) while the gain is positive -- a full derivation is given in this answer.
In fact, the gain and phase of oscillators can have pretty unusual shapes dissimilar to the smooth and generally downward-sloping plots seen for amplifiers (or amplifiers inadvertenly oscillating due to instability) -- the plot below is from the oscillator I am currently designing for a low-power radio application using a somewhat similar topology and an external resonator to establish the appropriate phase response and resonant behavior at oscillation:

Notice that there are two frequencies where the loop gain phase is zero -- one corresponding to a highly suppressed mode that does not oscillate, and one corresponding to the oscillatory mode at my intended output frequency. The one that does oscillate is the one with the high gain, of course.
This means that you get a sine at the right frequency from anywhere and it'll start building up more and more until it reaches the limits of the circuit's ability to increase further (at which point the &quot;linear&quot; part of our linear time-invariant assumption breaks down and the sine is sustained at the same amplitude without further growth). This is where saturation, and hence gain control, may occur.
Where does that initial sine at the right frequency come from? The thermal noise that's present everywhere in the oscillator contains all possible frequencies, and even the tiniest wisps of energy at the right frequency will build up cycle after cycle, growing until checked by nonlinearities which bring the gain back to unity.
This is of course not the only way to view oscillator topologies. One other topology that's pretty common is the Pierce Oscillator topology used for the oscillator on board microcontrollers such as the Arduino -- it can be seen as an RLC circuit (i.e. a lossy resonator) in parallel with a negative impedance such that the combined result is an RLC circuit with a negative resistance, i.e. a negative loss, i.e. a self-sustaining and growing oscillation.
1 This is a simplification. For maximum accuracy, you have to consider input and output impedances (so you really need to inject both voltages and currents, and test with both open-circuits and short-circuits-to-ground). You also need to consider so-called bilateral feedback: Even though the amplifier &quot;points&quot; in a certain direction, it can pass signals in the reverse direction very slightly.