I can do about 6-8 first set depending how I am training, am in late 50s, weight 70kg. Taken a long time!!
I agree about the machine not being the best, and you need at least some of the training to be at lower reps and harder progressions anyway.
Look up Meghan Callaway on insta.
Do scapular pulls (I warm up with these), negatives, holds (eg hold at the top, then forehead to bar, then 90deg etc), inverted rows for volume with different grips, band assisted, lat pull downs, cable rows, grip strength, hollow body holds. Start with harder progressions and low reps/longer rests and finish with easier progressions/higher reps/short rests.
The machine may help but it is a different and very specific path to the bar and less whole body involvement too. And you can't change grip easily. Doing a range of exercises will help you progress to your first pull up but will also build strength and power.
(If your PT gave you that machine programme to progress to a pull up, I'd be questioning their calisthenics expertise tbh. I have instructors who win competitions and none of them have ever suggested that other than as rehab after injury to build back slowly. They may be great on the more traditional gym stuff tho.)
Keep going - you will get there!!