Have you tried Lumetri?
As long as you're using a more recent version, there's a white balance eye dropper and also an "Auto" button.
Auto, as you probably know is hit or miss. It all depends on how well your footage matches the reference footage the the auto feature uses.
I normally use the Lumetri white balance eye dropper, but it assumes the white you click it on is a specific shade of white, which is almost certainly different than the actual white in your scene. It's sure better than guessing, but not as good as balancing to a known hue, which is what a card let's you do in principle. I can imagine an app that let's you tell it there's a ColorChecker card in the frame, it finds it, notes all the colors, and balances the frame to match. That's what I'm looking for.
The color balance part of the Basic tab only aligns the gain ... the top of the three channels RGB. So if you have a color cast down into the shadows, well ... that won't fix it.
I frequently go to the Creative tab, and use the shadows/highlight tint wheels to match something in the mid-shadows across channels, and something in the mid-highs across channels. Takes very little time to get to pretty neutral.
But no, there's not a way to auto-match to a color checker. Or ... you could color correct a still for reference of a close-up of your color checker, put that at the first of the sequence. In every shot you've got a color-checker, set the CTI for the checker, program monitor to Comparison view, and hit the Apply Match button. Then copy the Lumetri settings (copy attributes/paste attributes) to other clips without the checker from the same scene.
That looks like what I was thinking of. Thanks. Know of any others?