Why isn't the BIOS' ROM chip made using CMOS technology?

You're mixing up implementation technology with colloquial terms for functionality.

CMOS - Complementary Metal Oxide Semiconductor - is a method of making logic and related circuitry using both N-Channel and P-Channel field effect transistors. One of its defining characteristics is extremely low static power consumption - power is almost only used when changing state. As a result, a CMOS static memory chip can retain its contents for years on a battery, and is a handy place to store semi-permanent information.

BIOS and related startup code has traditionally been stored in PROM or EPROM devices. EPROMs in the era of the IBM PC were typically made in pre-CMOS technologies like NMOS, but the critical distinction is that these were not typically writeable when installed in the computer, but only in a special programmer. Further while many contemporary systems and clones used EPROMs, the actual production versions of the IBM-PC used cheaper non-reprogrammable PROMs (pinouts were typically compatible in use).

The PC-AT design then added a battery backed CMOS memory for storing customizable settings, and also (perhaps in the same device) a persistent real time clock. This came to be colloquially referred to by end-users as CMOS, though it was of course only a particular usage enabled by a then rapidly spreading chip technology.

Today of course we no longer use parallel E/PROMs for BIOS, but rather use serial NOR flash and transfer the contents into faster RAM for execution. Modern FLASH chips are in fact made of CMOS-derived technologies. And they typically can be reprogrammed in circuit. It's really up to the system designer if they want to put semi-permanent configuration information in a battery backed RAM, or in reprogrammable flash - the end user or post-boot operating system may have little actual visibility into a distinction.

But even if flash is used for settings, there's still typically going to be a low power real time clock that keeps running on battery.

Back when the PC was first invented, most of the logic on it was power-hungry NMOS and TTL chips. CMOS was very new and the only circuits in the PC that used it were associated with things that needed to run on battery when the power was off, such as configuration RAM and the real-time clock.

Nowadays, nearly all of the logic is CMOS, including the power-hungry CPU and the flash EEPROM that holds the BIOS. So in a sense, your question is based on an invalid premise — the flash EEPROM is CMOS. However, for whatever reason, the term "CMOS" in a PC still refers to only the RAM and RTC functions.

If you're asking why the BIOS isn't stored in volatile RAM rather than nonvolatile flash, that's because batteries fail, and erasing the BIOS would effectively "brick" the computer, requiring specialized hardware to get it going again.

I think you are confusing two usages of the "CMOS" acronym. There are chips built entirely of the complementary MOS transistor technology. In fact almost all chips these days are built this way including much of the digital control circuitry on a Flash chip.

The other usage of CMOS has persisted in the PC industry since the earliest days to refer to the chip that stored some settings and the real time clock with an attached battery. Back in those days (early '80s) the bulk of the large chips in a PC were NMOS technology and the miscellaneous logic chips were bipolar transistor logic (TTL, LSTTL etc). The only CMOS chip used was the RTC chip and so became known as "the CMOS".

These days the RTC is no longer a separate chip on a PC architecture device. Instead it is built right into the motherboard chipset (which incidentally is produced with complementary MOS circuitry). It is rare that any BIOS these days uses the legacy battery backed RAM in the RTC part of the chipset to store settings. Instead the BIOS uses a few pages of the SPI Flash storage device to save its settings in a non-volatile way. As such there are minimal settings that get lost in a PC when the battery runs down or is disconnected. The minimal settings that are lost on battery power removal are certain ones that control the power up and reset behavior of the chipset and are in fact not even stored in battery powered RAM cells but instead in special low power flipflop latches that are powered from the coin cell battery.