How 3-capacitor sine generator works really?

Hi Friends!
Schematic / simulation in "Falstad online simulator":
https://tinyurl.com/23hcg8np
This is probably very old and widely known schematic of single-transistor
generator which
requires no inductance, but instead uses three capacitors - actually it
seems a chain
of high-pass single-stage filters with transistor serving as feedback from
output to input.
One can find it, for example, in classic stylophone schematic (the part
creating low-frequency
oscillations for "vibratto" effect).
I teach it to my pupils for years probably and I always thought I less or
more understood what
is happening inside - each filter stage gives shift in phase and hence when
amplifying feedback
is added there happen harmonic oscillations.
However on the schematic given above I added 4 scopes over the length of
the filter (potentials
at the points A, B, C, D according to labels - here A and B are potentials
at points between capacitors, C is at the base and D at collector) - I
suddenly found that intermediate voltages are
not pretty harmonic! They could be distorted by the current drawn into
transistor base though. And
I'm not sure the output is exactly sine now. Though probably it is a matter
of adding some resistor  to improve input impedance of transistor cascade?
Regretfully I can't find any thorough explanation of the schematic
(probably due to keywords being too general and I don't know if this design
has fancy proper name). So I would be grateful either
for links or for verbal clarifications.
--
to email me substitute github with gmail please