Sinclair didn't always use the Z80 for its computers. The MK14 computer, sold in kit form (like the ZX80 was), used a National Semiconductor INS8060.
The ZX range of home computers have a video display hardware that is very closely tied to the architecture of the Z80.
On the two first models, ZX80 and ZX81, the video display hardware was kept to a minimum, yet allowing a decent resolution of 256x192 pixels and the ability to move the screen memory back and forward in the memory space, although it was not pixel addressable.
To accomplish this, the Z80 played an important role as RAM address generator for the video hardware (resembling in some way what the Motorola 6845 CRTC chip does for other microcomputers).
When it is time to read a TV scanline from memory, the Z80 is directed to "execute" (i.e. fetch instructions) from the memory address where video data is stored. The Z80 does not actually execute the instruction read. Instead, the video hardware peeks the data bus and retrieves whatever data the memory put there for the Z80. Just after that, and before the second clock cycle of the fetch bus cycle is ended, the data bus is forced to 00000000 so the Z80 ends up reading a NOP instruction, which does nothing and goes for the next instruction at the next address.
This way, the video hardware doesn't have to implement an address counter, logic to access memory and logic to perform bus contention, as it is the Z80 which is actually taking part in the video generation process.
The ZX81 is just a ZX80 with all the video hardware encapsulated in a ULA chip, plus some other features such as the "slow circuit", which was an improvement over the ZX80 video circuit, designed to avoid flickering while the user was inputting data.
The ZX Spectrum only shares some visual resemblance with the ZX81 video output, but it is actually a totally different beast. It could have been designed around a different CPU, but Sinclair had already a BASIC interpreter written for the Z80, and so the Spectrum ROM inherited much code from its ancestors, including some routines that shouldn't be in the Spectrum ROM because they made sense only in a ZX81.
Summarizing: the ZX80 and ZX81 relied on the Z80 way of performing a memory bus cycle to use it as a DMA engine. The Spectrum used a Z80 because it shared a lot of ROM code with previous models.