The Ebers-Moll equations, published in Dec 1954, describe a bjt as 2 current controlled current sources. Although you can derive Ic as a function of Vbe, it is also true that Ic is related to Ib, as well as Ie.
1) Ic = beta*Ib
2) Ic = alpha*Ies*(exp((Vbe/Vt) - 1)
3) Ic = alpha*Ie.
In equation 1), forcing a specific value of Ib results in Ic per 1). But this approach is avoided most of the time due to beta dependency. It is hard to predict beta with differing speciman & temp. In 2), Ies varies with speciman, hugely with temp, & a bjt driven w/ a voltage source across Vbe is thermally unstable. A Vbe connected to a source, results in Ic, some power, a temp rise, then Ies increases due to elevated temp, Ic increases, temp increases, etc. Thermal runaway means that you can never "voltage control" a bjt.
In eqn 3), alpha is very stable in value. Controlling Ie results in a very predictable Ic. It is thermally stable & predictable. Ic is ultimately controlled by Ie, not Vbe.
To see this, take a bjt w/ a very thick base region. A Vbe of 0.7V is set up, & Ie = 10 mA, which also equals Ib. Because the base is very thick, alpha=0, & all electrons emitted are recombined in the base, so Ic = 0. "Transistor action" does not occur until electrons emitted from the emitter are "collected" by the collector. Although Vbe=0.7V, there is no Ic.
What controls the number of electrons in the collector is the number emitted from the emitter. If 1010 electrons are emitted in 1.0 picosecond, then the Ic value can never exceed 1010 e-/psec, regardless of Vbe. The emitter emission literally controls the collector collection.