My handwriting is also awful, but what the parent suggested isn't at all incompatible with taking typed notes: I would type a first draft while reading the material before lecture, add during the lecture, then finish the final draft afterwards. I regret I didn't learn this method of note-taking earlier, and it works for both hand-written and typed. It also doesn't take that much time, but the last step of reviewing and cleanup is critical for actually cementing my understanding of the material.

Another possible method would be taking written notes during lectures, then typesetting them afterwards. However, I found it was preferable to revise my notes soon after the lecture, at least 1-2 days following, while my memory was still fresh.