Need help making STM32F303 board
In addition to the answers already given, I would like to point one one more issue that wasn't mentioned.
The 8 MHz HSE clock circuit is connected to the ST-Link and then routed via the MCO line to the STM32F303. If you remove the ST-Link, you also lose the HSE clock.
Without the HSE clock, you can still achieve a system clock of 72 MHz. But there are other restrictions, e.g. you cannot use USB without HSE.
Therefore, it might make sense to add an 8 MHz crystal and the required capacitors to your design.