STM32F303 code gets stuck in SysTick Exception Handler
The ST HAL requires the systick for operation.
It increments the system tick you can use via HAL_GetTick(), it is incremented in the SysTick_Handler() via HAL_IncTick();.
It will be configured by HAL_SYSTICK_Config somewhere in the HAL.

You don't get stuck there. It just gets called every 1 ms.

If you do get stuck in the systick handler, there could be something wrong with the vector table. Eg: the address for SysTick_Handler at the position for something else.

Check the IABR registers when halted.
4.2.6 Interrupt Active Bit Registers.