please explain what do you mean by “power GPIO pins”?
CORE2 and CORE2-ROS can’t be powered via USB port - please read this section of manual to find out why: https://husarion.com/core2/manuals/core2/#hardware-controlling-servo-power-supply
It will be partially powered and it will cause that LEDs on board and the microcontroller will be powered, but it is not the proper way to supply the power. It can be done only for checking if SBC works, but not permanently!
When you tried to connect the power from USB port of your laptop, the red LED probably turned off after 10s because the SBC have started up and draw more current at this particular moment.
When you tried to connect Tinker Board and CORE2 together and power the CORE2 via its power connector, and it didn’t work as well, it is possible that your power supply was still not efficient enough.
I believe that when you find the proper power supply, everything will work fine