why not just scan my GE card when I show up and decide on whether I can use PreCheck instead of asking the airline to submit data (via BP) to then figure out if I would be granted PreCheck or not?
Because the airline does not make the decision. TSA makes the decision. Therefore, airlines must submit the KTN to TSA in order for TSA to let the airline know whether it is okay for Pre-ck to show on the BP. As mentioned earlier, having a membership in one of the Trusted Traveler Programs only makes one eligible to receive Pre-check screening - it does not guarantee actually getting it.
Many things can happen between a) being approved for a TTP and receiving a TTP card and b) when printing a BP and showing up to board a flight. So just showing a card at the screening checkpoint is not enough.
But to your question "Why cant I just show up and then they figure out if I get Pre?" That can and does essentially happen. If your BP does not show Pre-ck because your KTN was not entered or there is some data discrepancy, you can go to the ticket counter and have the airline update the reservation with the KTN or correct info. The databases are updated frequently but not in real time so it may take a little while (as much 20 minutes) for the databases to update and Pre-ck to be approved for the BP.
But the process is much smoother when the KTN is sent to TSA long before the pax checks-in.