Hi @backflip.
It is a bit of a stretch to call that a "programmer". This is simply an adapter that produces a virtual COM port on your computer via the USB connection and then converts that interface to TTL UART communication. So it can be used as a "bridge" between the computer and a microcontroller (or any other device that uses UART communication).
Usually when we talk about a "programmer" device, we are referring to something that is capable of directly writing binaries to the microcontroller's flash memory (e.g., the STLINK-V3).
The "STC" thing you got is definitely a useful tool for communication with your STM32G030C8T6 dev board, but you aren't going to be able to use it to flash a bootloader binary to the microcontroller, and you also won't be able to use it to upload sketch programs to the microcontroller unless the microcontroller already has a suitable bootloader installed.
It is just some random branding the Chinese manufacturer screen printed onto the case. You will find the same exact hardware sold on the various online marketplaces with various different branding. Under the hood it is just a WCH CH340 chip with the basic support components. So don't poison your searches by adding irrelevant keywords like "STC".
Something that is very confusing is that the Chinese also make ST-LINK/V2 clones (which are truly "programmers") in the exact same form factor as the "STC" USB to serial adapter you got. Sometimes the sellers of the USB to serial adapters even add terms like "ST-LINK" to the title or description of the product listing, even though the USB to serial adapters are completely different hardware than an ST-LINK. So you might find tutorials on the Internet where people are using something that looks almost exactly like the adapter you have to successfully program or debug an STM32 board.
I can tell you that the reasonably priced STLINK-V3MINIE works perfectly as a programmer when used with the "STM32 MCU based boards" Arduino boards platform (AKA "STM32duino").
I have only used it with the more common STM32F103C8 dev board ("Blue Pill") that I have on hand, so I can't say for certain that it will work with your STM32G030C8T6 board, but I don't have any reason to think it wouldn't work.
I use the Tools > Upload method > STM32CubeProgrammer (SWD) configuration.
I think Tools > Upload method > OpenOCD STLink (SWD) would also work, but unfortunately that configuration can't be used on Windows computers, only on Linux or macOS machines.
The "STM32 MCU based boards" platform (AKA "STM32duino") is maintained by manufacturer of the STM32 microcontrollers, STMicroelectronics.
That forum isn't super busy, but it certainly isn't accurate to say "not active anymore":
It is absolutely worth the effort. Many members of the Arduino community use Arduino IDE and the "STM32 MCU based boards" platform with their STM32-based boards and have created a tremendous number of amazing projects.
Of course there are excellent alternatives to Arduino IDE, but if you want to use Arduino IDE with your board then you can definitely accomplish that if you just put in the effort to get the right hardware sorted out.
If you want to go on easy mode, get one of the NUCLEO boards manufactured by STMicroelectronics. These have a built in STLINK-V3 which acts as a programmer, USB to serial interface, and debug probe. They are available for a very reasonable price.
Alternatively, you might take a look at Arduino's own GIGA R1 WiFi board. This board uses the STM32H747XI microcontroller, which has significantly more resources. The Arduino company provides Arduino IDE support for this board through the official "Arduino Mbed OS Giga Boards" platform. When using the GIGA R1 WiFi, you just connect it to your computer with a USB cable and you are all set to upload sketches and communicate over serial without any need to worry about connecting some external programmer to the board.