You don't need a STLink, you can use the bootloader, but if you use HID there is no way for the IDE to reset the board back into the bootloader, as it normally sends a magic sequence via USB Serial to the sketch, which causes it to reset and hence run the bootloader
So in HID mode you'd need to manually press reset just before each upload, to get it to go back into the bootloader.
And to load the bootloader you need a USB to Serial adaptor of some kind, (you can even use an old Arduino Uno as a USB serial just to install the bootloader)
STLink can be handly however and they are dirt cheap on eBay, (I have at least 3 of them)
If you use STLink, you won't need to press reset before uploading, as STLink directly programs the STM32 via its programming pins SWDIO and SWCLK
Re: Using an F103C
You may be able to use one, but I think the multiplexing is going to be a pain.
You'd be better off buying a bigger board
BTW. I don't know of anyone that has used 6 Analog inputs. I know a lot of people have used 2 Analog inputs and even used DMA to read them in very quickly (which is not what you need).
i.e its just a note of caution, as although it should work with multiple inputs its not something thats been extensively tested.
Id just go for one of the VET boards likehttps://www.aliexpress.com/item/STM32F1 ... 80d84fc483
The one I have is this one.https://www.aliexpress.com/item/1PCS-ST ... 80d84fc483
Loads of pins. Far more than I've ever needed to use