Thank you, Jianyu, for confirmation that I should be able to rely on Config Tool to set up my microcontroller.
Then I guess the next question is, is there something in the config tool I am setting up wrong? Here is a screenshot of my clock config page:

So based on your and Mark's comments, because I have Core clock set to 48MHz from the HIRC then my SysTick timer should be using that clock because we confirmed SYSTICK_CSR (register at address 0xe000e010 in the Cortex M0+ core) bit 2 is set. That means my SysTick timer clock should be set correctly. Is there another reason why my SysTick timer might not work on my board but does on the FRDM board?
Here is the timer_kinetis code I imported from FRDM MKL02 board, because the MKL03 board didn't come with a timer. It works on the FRDM MKL03 board (!!) but not on my board.

I have left all the parameters default from the freshly download MKL03Z32CAF4 SDK from online and use the Config tool to set my pins and clock. I feel like I am missing something obvious but I don't know enough to make changes on my own.