MarkT:
Beta is nothing to do with this, beta is the small signal current gain in linear operation, which does not
apply to switching (where the transistor is either saturated or cut-off, not in the forward region).In saturation you need the base current to be about 5 to 10% of the collector current for a BJT.
For that amount of stall current a MOSFET or darlington are needed, a single BJT cannot do the job.
Beta (or "Forced Beta") is a way of expressing the set point when designing with a transistor. Beta is merely the ratio of IC to IB, and the way to put a transistor in saturation is to force the beta to a smaller value than the specified beta (i.e. the HFE in the datasheet -- or the beta that results in Linear operation). A transistor is an analog device. There, really, is no such thing as an active "switch". There is merely a device that can be set to various degrees of control. For bipolar transistors, it's current controlled by current. For a MOSFET, it resistance controlled by voltage. Just as a MOSFET can never have a channel resistance of zero, a transistor can never have a Collector-Emitter with no voltage drop when controlling current. There is the ideal switch (0Ω when closed, ∞Ω when open), then there is the reality: analog devices being made to function like a switch -- but, it's always imperfect -- no transistor can ever reach the ideal model.
One example of the importance of managing beta is making transistor logic transition faster. By managing beta, to keep the "on" state at the point between active and saturation -- i.e. never allowing the transistor to reach the saturation state -- makes the "switch" faster, because the more a bipolar transistor moves into the saturation region, the more a charge develops at the base -- which must be discharged before the transistor can even begin to transition towards "off". But, the further the transistor moves into the active region, the more current is allowed to pass through the Collector-Emitter junction. So, it becomes a balance between switching speed, and current demand.
In other words, there really is no such thing as "all the way on" -- all the way on would be a VCE of 0.000...V
AND, there is no such thing as all the way off, either. There's always some leakage current.
There are only degrees of "off" and degrees of "on" -- because an Analog device is being used to mimic a switching function.
With a Bipolar Transistor, it's all about current ratios. And Beta is a measure of the target ratio at play, for a particular application. For a switch, the beta is in the lower ranges -- typically 10. For, applications in the active range, the beta is higher. It's like an Op-Amp. The beta rating of a transistor is like the open-loop gain of an op-amp. And the functional beta of a transistor circuit is is like the gain in an op-amp circuit, programmed by the resistor network.