Always hard to guess why something simple hasn't been done.
In case of the Spectrum it might have been to save on component price. The modulator used is the same as for the ZX81, so Sinclair was for sure already at the optimum price point. On the down side it's a simple video modulator (*1) without a mixing stage for sound, after all, in its quest of maximum reduction the ZX80/81 did skip sound as well. Next to all other home computers (*2) of that era had sound and used the TV set. Without sound not only the sound circuit could be saved (*3), but more important money was saved on the modulator.
Sinclair could have voted to use a modulator capable of adding sound to the Spectrum, but buying them would have started at a higher price point due lower (start) volume - or to add a cheap speaker on board and share buying power for components with ZX-81 production.
*1 - In fact it's so simple, that they already had to add quite some analogue gears to make it work.
*2 - TI 99/4, Atari 400/800, Commodore VIC, C64, C16, Tandy CoCo, MSX (pick your favorite), eventually the majority of home computers did. The Apple II is a notable exception, but then again, it's a whole generation before home computers came.
*3 - Not much, as this could have been done with a single signal pin and maybe a driver transistor.