How to find out if a binary number is zero

There's simply no way around ORing all the bits, as unsatisfying as that may seem. However, you are not restricted to two input gates in silicon either. You can build a 4-input NOR gate in CMOS logic by putting 4 series p-type transistors in the pullup network and 4 parallel n-type transistors in the pulldown network. That reduces the depth of your tree topology and therefore your propagation delay. You can only take that theory so far though before the cumulative voltage drop across the series transistors makes the pull-up not pull-up enough to be a "1"... four is a good rule of thumb if I remember correctly.

The logic function is the NOR gate. That is the simplest logic function that exists.

The typical solution with 8 bit machines was that the ALU would produce a number of 'flag' bits that would represent the outcome of the most recent operation. While it would be possible to have any number of flag bits around (i.e., you could have a 'Z' flag for every register in your CPU), it's usually the thing you've just computed that you're most interesting in, so it makes a certain degree of sense to do it that way.

Some of those old CPUs would automatically set flag bits for almost every data move, while others would require you to stick a specific 'compare' instruction in your code if you just suddenly need to know if a certain register was zero. And whether you provide a zero check for every register or just for what's just been computed, there really is no simpler way to check for "is this word zero" than to just OR all the bits together.