Strength / Weight Training

Yes mainly. You can do assisted pulls with bands and slow negatives if you jump up. Lat pull downs can help a bit, but nothing is the same as the compound from pull ups.
I used to hate them but persevered and actually enjoy them now

I found volume really useful for building these. Just throw in an easy superset (stop the second it starts to get slow) straight after basically anything thats not already a pulling movement. Even if it's just 2-3 reps, it adds up to 30+ a session real quick.

fat and weak.

All of us

For rotator cuff see about getting some rings and doing pull ups on them, allowing joints to move as they desire in a pull up should make it less painful.

and yes the long and short of it is "do more pull ups" with a side dish of "get stronger at pulling in general".

If you can do 4 or 5 then yeah best method will be just doing more pull ups probably. You could try greasing the groove - doing extremely submaximal sets (like, 1 pull up) many times a day. Like, 10-20 times, but spread throughout the day.

I found success in pyramid sets. Do a rep, rest for a short period, do 2 reps, rest, etc, until you fail a set, and then do back down. e.g. 1-2-3-4-5-6(fail@5)-5-4-3-2-1, and for the sets going back down add in negatives when you can't complete a rep.

If you want to really prio them you could look at the russian fighter pull up program https://www.strongfirst.com/the-fighter-pullup-program-revisited/

Agree that bands / negatives etc are best but there are a few other things you can do that might help ease the load. Some ideas: https://meghancallawayfitness.com/my-blog/learn-pull-ups-with-these-8-exercises/

Also, the little and often method (the calisthenics crew sometimes call it "greasing the groove") is definitely good for increasing volume, but obvs be careful if you've got rotator issues. Pull up bar across a door frame etc make this easy to do.

Do you have to do rotator specific physio (and if not, worth adding some to your workout)? I snapped the back of my shoulder joint off years ago, and have memories of having to do interminable number of "pull the exercise band across your body" variants to build the muscles back up.

Edit: just seen @zooeyzooey's post - what he said!

Same.
This year I just do 2 pull ups in between every exercise I do at the racks. Will generally mean I accumulate 40+ each time. I went from like 5 in a row to 10. Not huge progress but it has helped and is basically no extra effort on my part