Is BJT a Voltage controlled device or Current controlled device?
The answer depends on your perspective.
A physicist might say that the fundamental action in a BJT is that an electric field across the base-emitter junction decreases the width of the depletion zone. It is this electric field, measured in volts, that controls the movement of charge carriers. Therefore the BJT is voltage controlled.
An electronic engineer might say that the most useful model for a specific circuit design is the current-amplifier model.
So called "common emitter current gain" is a range not a constant. Good designs don't depend on it.
Short answer: the Ebers-Moll model gives a relationship between the collector current and the base-emitter voltage. So you can view the base-emitter voltage as being controlled by the collector current or as the collector current being controlled by the base-emitter voltage.
Many people make the incorrect claim that there is a useful relationship between the base current and the collector current, and thus mistakenly claim that a transistor is a "current controlled current source." A transistor is not a current-controlled current source.
Long answer:
The confusion about whether a BJT is current controlled or voltage controlled comes from two sources. The first is that the equations we use in describing electric circuits are not definitions of one variable in terms of several others. Rather they are describing a constraint between several variables. Take Ohm's law: \$V = IR\$. This is not a definition of voltage. Nor is \$I=V/R\$ a definition of current or \$R=V/I\$ a definition of resistance. Rather it says that in any circuit (involving an ohmic device) this equality will always hold. No matter how we change the current, the voltage will always stay proportional to the current. No matter how we change the voltage the current will always stay proportional to the voltage. (True story: I once received a resume from a gentleman who listed as one of his qualifications that he knew, and could use, Ohm's law "in all three forms.")
The most important constraints in describing how a transistor works within a circuit are the Schockley diode equations used in the Ebers-Moll model. In the active mode this results in the constraint that: $$ I_E = I_{ES} (e^{V_{BE}/V_T} - 1) $$
where \$I_{ES}\$ is a constant that describes the transistor, and \$V_T\$ is the thermal voltage (about 26mV at room temperature). So this is describing a relationship (constraint) between the emitter current, \$I_E\$, and the voltage between base and emitter, \$V_{BE}\$. Yes, the current is on the left hand side and the voltage is on the right hand side, but this is only because the \$-1\$ makes it a little difficult to write the other way around. In fact, when \$e^{V_{BE}/V_T}\gg 1\$ it is sometimes useful to write \$V_{BE}=\frac{1}{V_T} \log(I_E/I_{ES})\$.
Nonetheless, the physics behind the Ebers-Moll model, is usually thought of the way that @RedGrittyBrick describes it: the voltage between base and emitter controls the current of minority carriers into the base (given the relative dopings of emitter and base).
The second source of confusion comes from another statement that people make about transistors that is just completely false. This is a statement that a transistor has a well defined "common-emitter current gain", or \$h_{FE}\$. I will write this really large so people don't miss it:
A transistor has no (well defined) common-emitter current gain.
It is definitely the case that there is a flaw in bipolar junction transistors where there is always a leakage current through the base, but the leakage current is not well defined between a pair of the same type of transistors, nor is there any simple linear relationship that describes the base current in terms of the emitter current in a specific transistor. The current through the base is caused by a number of factors, like relative doping levels of the base and emitter and the width of the base, that are difficult to control during manufacturing. Let us take a look at the datasheet for the Fairchild PN2222. You will see that \$h_{FE}\$ is given as a range. It is somewhere between 100 and 300 (a factor of 3 difference !) when the collector current is 150mA. But \$h_{FE}\$ is not less than 35 when \$I_C\$ is at 0.1mA. Another factor of 3 different! So \$h_{FE}\$ is not like the measured resistance of a resitor. \$h_{FE}\$ is not a constant and is not a useful description of the transistor's gain.
When designing an amplifier the only thing you use \$h_{FE}\$ for is to decide whether the transistor's leakage current will be bearable for you or not. If the \$h_{FE}\$ is too low for your use case you'll either have to choose a different (probably more expensive) transistor, or replace the single transistor with a Darlington pair.
Now I will write this really large again so people don't miss it:
A good design never depends on \$\beta\$ (\$h_{FE}\$) having a particular value.
Try the following Wilson current mirror to see how you build a current-controlled current-source. Q3 is specifically included to reduce the dependence on \$\beta\$. I encourage you to change all the 2N3094s to 2N3055s (or to any of the other transistors that has a different \$\beta\$ than the 2N3094) to see that the output current is always about 2x the input current.
simulate this circuit – Schematic created using CircuitLab
Please enjoy an answer in the form of rhetorical questions:
Does a current through a resistor cause a voltage across it, or does a voltage across a resistor cause a current?
Does a motor spin your car's wheels, or do the wheel resist the motor's spinning? Does shifting to a lower gear increase the engine torque at the wheels or does it reduce the resistance from the wheels at the engine?
Does a college education make your expected salary higher, or are successful people more likely to attend a college?
Did the decline in pirate population cause global warming or did global warming kill off the pirates?
Point is, an explanation of how a BJT is "controlled" is a fallacious attempt to assign a cause and effect relationship between voltage and current when really there's only a correlation (except unlike pirates and global warming, the correlation is very strong and observable). We can think of this correlation as cause and effect when it suits our needs, but it's only a model to help our reasoning in a particular case. Both explanations (a BJT is voltage controlled / current controlled) are valid models, each appropriate in a different context.