You have to turn that option on in the Screenshot app's Option menu. SHIFT+COMMAND+5, then go to Options and check the option at the bottom of the Screenshot menu:
"Show Mouse Pointer"
Here is a screenshot showing the mouse pointer after turning the option on:
No, my mouse pointer is visible when taking a screenshot after enabling that option. I wonder if you're experiencing a bug. Which version of macOS are you running? I'm using macOS Catalina 10.15.1 on a 2019 MacBook Air (base model). Just a wild guess, but have you tried disabling the option and then re-enabling it?