Ohm law tells me that if I don't have any resistance, the current will be infinite.
And that is correct, in practice the current will be limited by the zenerdiode's series resistance and/or how much current the transistor will supply to the zenerdiode (which depens on how you control the transistor's Vbe). But even then the current will be too high and will potentially destroy some components. So you do need some form of controlled current limiting.
But in this case I can limit the current by controlling the base current of Q1. And that is also correct BUT you would need to know the exact beta (Ic/Ib) of this transistor. Beta varies a lot over transistors even when they are the same type. So although in theory controlling the current through Ib and beta works, it is highly unpredictable so a practically useless method.
You could make a current mirror to create a Vbe for transistor Q1 but then also, you would need a fairly identical transistor. This is how this can be implemented on a chip because there the transistors are created much more equal. With discrete transistors there is a trick, you can add emitter resistors that drop around 0.5 V or so. That will make a usable current mirror for this purpose.
But it is much easier to use the transistor to only switch on/off the current (do use a base resistor to limit the base current btw) and use a resistor in series with the zener diode to limit the current.