It seems a little inconvenient to require acceptance that empty products equal 1, since that is also slightly subtle and deserving of its own explanation of mathematical terminology.
Of course, I generally hear the fundamental theorem of arithmetic phrased as “every integer greater than one…” which is making its own little special case for the number 1.
>It seems a little inconvenient to require acceptance that empty products equal 1
Only the contrary: it is extremely inconvenient to not allow the product of an empty sequence of numbers to equal 1. The sum of an empty sequence is 0. The Baz of an empty sequence of numbers, for any monoid Baz, is the identity element of that monoid. Any other convention is going to be very painful and full of its own exceptions.
There are no exceptions to any rules here. 1 is not prime. Every positive integer can be expressed as the unique product of powers of primes. 1's expression is [], or 0000..., or ∅.
Any convention comes with the inconvenience of definition and explanation. So to call the convention that the empty product equals 1 based on that alone seems a bit unfair. The reason the mathematical community has adopted this convention is because it makes a lot of proofs and theorems a bit easier to state. So yes, you lose a bit of convenience in one spot, and gain a bit in a whole bunch of spots.
And note that this convention is not at all required for the point I'm making regarding prime numbers. As you say yourself, restrict the theorem to integers greater than 1, and you can forget about empty products (and it is still easier to state if 1 is not prime (which it isn't)).
Of course, I generally hear the fundamental theorem of arithmetic phrased as “every integer greater than one…” which is making its own little special case for the number 1.