How bad an idea would it be to just let my employer know what's going on?

Extremely bad. People have been fired or denied promotion because of this. Don't even tell any of your colleagues.

I am not discussing the legal aspects of this, but you will probably be perceived as not worth investing in the long term. Imagine that your interview fails and you decide to stay. Your current employer is not going to trust you with anything important anymore, because they will be expecting you to leave soon anyway.

Okay, this may sound irrational, because you are not your employer's slave, and technically you are (any anyone else is) free to leave sooner or later. But people still make estimates. It is in your best interest to pretend to be a loyal and motivated employee, until the day you are 100% ready to leave.

It feels deceitful

This is part of the human nature; what we have evolved to do. Even your dislike for deceit is part of the deceit mechanism. If you unilaterally decide to stop playing the game, it most likely means you lose.

There is probably an article by Robin Hanson about how LinkedIn helps us to get in contact with new job offers while maintaining plausible deniability, which is what makes it so popular, but I can't find the link now.