I think I understood the explanation in general, but there are a few details I want to explore. I'm aware of the bad design. I mean, not the design, but the values for the resistors, as the design is a common one, I guess, nothing special about it.Perhaps you are interested in the reason for this difference.
The reason is in your design which is a bad one. Let me explain:
Normally, the resistive voltage divider at the base is selected to be as low-resistive as possible because each good design should provide the required base voltage - independend on the pretty large tolerances of the current gain B (resp. beta).
(Rule of thumb: The base current should be max. 10% of the current through the divider chain).
The lower limit for both resistors is set by the allowed power consumption as well as the input resistance of the whole stage.
In your design, we see two resistors - each of 1 MegOhm.
As a result, the voltage at the base is strongly dependent on the base current which is different for the various transistors.
Example: I have simulated your circuit with the given values and a BC108 macro model.
The emitter current was 1.59mA.
For a better design with a low-resistive voltage divider and a suitable emitter resistor you will have an emitter current which will be much less dependent on the selected transistor type.
A few things I would like to discuss:
I don't understand when you say that the base current is different from transistor to transistor. I might be misinterpreting the purpose of the voltage divider there but, isn't it to take advantage of only have a single power supply so that we can have current at the base? I mean, if there were 2 power supplies, we could use one at the base and another at the collector but when we only have one, we use a voltage divider there so that we can use the single power supply to feed both terminals, right? So, if this is the purpose of the voltage divider, then we control the current going into the transistor base, therefore, not depending from one transistor to another.
Lastly, I wasn't expecting any difference at all, because if @Jony130 used simply math equations, then, results should be exactly the same.
I'm pretty sure no one will follow the steps I attached but, if no errors manipilating the variables, I should get the same values he found.
I'll just plug smaller values for R1 and R2 and see the difference in \( \displaystyle{I_{b}}\).