Can't speak to the whole routine but for pullups I'd suggest negatives first and then banded pullups. Some gyms even have weight assisted pull up machines
I'm betting 50$ that whomever is telling you that can't properly explain what they mean by that.
You can safely ignore people that talks about "activating the core" as they generally don't got a clue.
If you want to be able to learn pull-ups, doing machine assisted pull-ups are perfectly fine.
Negatives are just starting at the top of the bar and trying to slow how quickly you drop down. If you can't do that, then there probably some other fitness priorities you should be focusing on instead.
You can always do negatives. Put a bar around shoulder height, stand straight, grab it, and slowly lift your legs. You will start to slide down because you can't hold your bodyweight without putting your legs on the ground. So you'll slide down. Congratulations, you did a negative pull-up.
If you can't do negatives safely, work on just dead hangs with core activation, and move to scapular pulls just activating the lats.
You can also try inverted rows, where you take something like a bar and do "pull ups" or rows with your feet on the ground and pulling your chest up. Search those up; it's not an exact one to one, but will help developing strength with a kind of similar motion.
as a side note, make sure your checking your posture and shoulder flexibility as you do it. many people do them with an arched back. you should be able to hang straight down through the whole motion while doing them. that's the part i struggle the most with.
set your back flat against the wall and slowly raise your arms up from the side keeping them flat against the wall. if your back or arms need to lift off the wall at all as you reach all the way up you should work on your flexibility as well.
There's no problem with doing pull-ups with an arch. It's just a different variation that uses a different set of muscles. Most people find it easier to pull with an arched back.
One way to start is to have someone or some thing hold your legs up while you do them, assisted pull-ups. Each time after, try to do as many solo pull-ups as possible, followed by assisted pull-ups.
Continue this routine until you are able to do them comfortably up to a specified amount.
I used to try to improve my pull-ups for a long time with not particularly good results. I switched to doing chin-ups, and I'm starting to see actual improvements in lat strength now.
Then again, I also started doing deadlifts and barbell/dumbbell rows so that might have had something to do with it as well