I finally managed to make AJAX form work and be able to include the form in any template. Everything seemed to be working, until I added reCaptcha (v3) field + process part. Form is being submitted and I can see for a fraction of a second HTML part replaced with errors, but then whole page gets redirected to the plain form template (which should be returned only by AJAX call).
Is there something I'm missing? I don't have for now how to test it on fresh installation (as soon I can, I'll try to submit a recipe to docs Cookbook on how to achieve this form as a widget behavior) :(