Hey @cyberdave, thanks for your response. This is the Webflow site where I'm setting this up. Here's a screenshot of Google's setup instructions:
The client-side integration is easy as pie. I've put the script in the head via Custom Code in site settings and the div in an embed block in the form. It's showing up and you can click it, but (1) you can submit the form without even touching the widget, and (2) I haven't set up the POST request to communicate with the URL they provide. (Though I did write my first few lines of PHP yesterday.)
What's losing me is the serve-side integration. Their instructions for server-side integration are here.
Also, there's a Google official Github repository for this service that provides two php files. I'm in the process of figuring out what exactly they're for and how to use them.
I could be completely off here because I'm new to this backend stuff, but I think that in order to make this work with Webflow, one would have to write a form-handler PHP file and link to that file in the Action field of the form settings. If I'm not mistaken, that would override the Webflow defaults and you'd have to do the rest of the form handling in that PHP file (emailing results to the site admin). Which would be a lot more hassle. It would be great if Webflow can still handle the form the way it does, but require the user to pass the captcha test.
Thanks for reading all the way down here. Your time and effort are much appreciated!