I've done some blinking led projects with STM32 but never even connected the reset line. Just GND, Power SWDIO and SWCLK, though the nice and small ST-Link V2 clones. And reprogramming the led does change it's flash rate, so I know it works. Program is also in flash because it starts flashing again after a power cycle.
STM Cube did complain about old firmware, so I let it update one of my ST-Link clones, and even though it does not have an STM processor inside, it still works. You have to be aware that these clones with an aluminium tube and USB -> 10pin IDC are not all the same. Quite a shame that ST itself only has much bigger programmers, and also includes them in their nucleo boards. It makes those boards bigger and more expensive.
For my own projects I still strongly prefer to buy genuie STM parts (That is, if it's possible to obtain them at all), as I have no interest in subtle incompatibilities between the 8 or so different Chinese clones.