all waves should have a frequency, and thus a pitch
No. In order to define a frequency of the wave, there must be periodicity.
The simplest to describe pitched sound is a sine wave – it keeps repeating over and over. Most instruments generate waves that can be described as a sum of several (or several tens) of sine waves, with some fundamental frequency f, the next twice higher: 2f, another tripled: 3f and so on. The wave formed by the sum of them repeats with frequency f. (Sometimes some frequencies in this sequence are missing or are not heard, but the sum of them still repeats with frequency f).
Waves of unpitched sounds cannot be described by a sum of limited (small) number of sine waves with frequencies being multiples of a common fundamental. We can still mathematically try to answer the question: sum of what sine waves would form this wave?, but the answer is: infinite number of sine waves covering a continuous range of frequencies. Such waveform doesn't repeat.
In practice the distinction isn't binary. Most pitched instruments produce some unpitched sounds with continuous spectrum, like the sound of piano hammer hitting the string or hiss of air in a wind instrument. On the other hand while unpitched instruments produce a continuous range of frequencies, they may emphasize some subrange of it strongly enough that a pitch can be recognized; this way, generally considered unpitched drums still can be tuned to a given note.