I agree with your interpretation.
Regarding the current drawn from pin 3 :
at point A the current is at it's Max and at point B at it's Min.
The higher the current drawn the lower the voltage drop at the output of the 555.
The result is you have a ramping exponential-like output level instead of a "flat plateau one".
Illumination?
Not really, but the above is why you shouldn't use the output signal in a timing circuit of the 555
like in this "non-working" circuit !!!
Instead of charging the capacitor from a "regulated voltage source" you are practically charging it with an un-regulated one,and a timing error will occur.
You can do a few more tests:
1. Add R1=100k to the load(from the junction of the R-C to VCC (like in the original "non-working "circuit)
2. Use an R-C load with R=33K and a corresponding lower C.
3. Use only the 1k resistor as load.
Regarding the current drawn from pin 3 :
at point A the current is at it's Max and at point B at it's Min.
The higher the current drawn the lower the voltage drop at the output of the 555.
The result is you have a ramping exponential-like output level instead of a "flat plateau one".

Illumination?
Not really, but the above is why you shouldn't use the output signal in a timing circuit of the 555
like in this "non-working" circuit !!!
Instead of charging the capacitor from a "regulated voltage source" you are practically charging it with an un-regulated one,and a timing error will occur.
You can do a few more tests:
1. Add R1=100k to the load(from the junction of the R-C to VCC (like in the original "non-working "circuit)
2. Use an R-C load with R=33K and a corresponding lower C.
3. Use only the 1k resistor as load.