If you want to take notes during lectures, you need to learn how to summarize in your own words. You need to write the stuff down so you can remember what was in the lecture. Something you can summarize the whole sentence with one word. Notes are for you, not for other people to read.
The idea isn't to merely take down a record of what was said, but rather to integrate the concepts into your own understanding: What associations can be made with existing knowledge, can you demonstrate to yourself your understanding of an abstract by contriving examples of it, can you explain it to your grandma, do you have a clear model of its scope/boundaries/limits and of where it fits into its domain, do you know the why as well as the what, etc.
The part where we have to condense what we're hearing into something concise enough that we can keep up is a feature. For that reason I wouldn't really recommend transcribing the lecture and then making notes out of it later, either (ideally we should be reviewing and organizing the notes into a more structured / fleshed-out / pared-down (as needed) form later anyway, so it would just seem to add an extra step). I don't know what everyone's courses are like, though, so maybe that approach could be useful or necessary for someone.
(And of course the takeaway here is that, assuming my understanding of it is well-grounded, and provided there isn't truly something inherent to handwriting that makes it cognitively advantageous, typing should have the potential to be just as workable a medium.)
Even when just copying what the professor what writing on the blackboard I could barely keep up.
It’s hard to summarize those lectures on the fly, and it might be a mistake as you’ll miss some important steps.
I found the exact opposite. My math and physics lecturers were expecting us to write what they wrote on the blackboards or overhead projectors down. So they had prepared the lectures this way, they wrote at a decent speed so you had time, and you could think about the steps as you transposed it to the notebook.
I found many small mistakes the lecturer had made this way, forgotten a sign or forgotten a factor, as I would follow along the whole time and go "wait, how did he get this from that".
I wrote it all down, but I seldom had to study them afterwards. I would often just gloss over and I would recall it all.
This was in stark contrast to other and later classes where the university administration had pushed hard for slideshow-based lectures. There the slides moved so fast I didn't have time to take proper notes, or I had to sacrifice listening to the lecturer. I ended up stop taking notes and just listen, but the memory of that wasn't as good as the handwritten notes, not by a long shot.
I'm sure people have anecdotes of students furiously typing down every word of a lecture, but that's the same point for people handwriting/half stenographing fast enough to do the same. The attitude isn't bound to the medium.