I just wanted to try to get you to do and understand one thing right: the common emitter circuit.
The key to understanding is to stay with the subject, avoiding the introduction of side issues.
I set out a modest experiment, but without a voltmeter it's pointless.
(You have an ammeter, but no voltmeter?)
There's a big difference between VB, VBE (approx 0.7V), and VCE.
With regard to "specific beta", all design assumptions are, or should be, based on worst-case, h_fe/beta guaranteed Minimum.
In some production lot there would likely be little variation unit to unit, but from lot to lot, or manufacturer to manufacturer there's no telling.
You may find that your transistors do better than the guaranteed minimum, well and good, so you can utilise that overage.
Anyway, till we can agree to work on one specific circuit, really wring out the fundamentals, I can't see any point in going on.