Adding to previous comments, it is important to think of helping a child to sleep better as a journey, a process, not a right or wrong way.
It sounds like you've worked out some patterns with your baby
Now with this,
Earlier I just walked out the room after the 15th or so PU although she was still crying, which turned into a mantra cry when I left and she stopped. 5 mins later it was time to wake her from her nap. She was just about to fall asleep again! Does that negate our efforts if I don’t actually let her get back to sleep when she is learning to self-sooth?
I'm a bit confused, but I assume you were trying to get her to sleep, she settled enough with a mantra cry for you to leave the room & settle, but then you looked at the clock & it was time to feed her again. For me I'd say firstly - watch the clock less, if your goal is to have her sleeping better then let her get the sleep... as a parent who's BTDT (& struggled loads along the way). ALWAYS look at the big picture, much older example, but I think it clearly explains, if your goal is to get the child to a bath as efficiently as possible & it works by piggybacking them to the bathroom every night... you are achieving your goal, if you have to drag them kicking & screaming because you think they should do it happily your way, then your goal isn't to get them to have a bath, your goal is to get them to go to the bath "your way"... 2 different goals, KWIM. So with the sleep, you kind of need to look at if your goal is to achieve more independent sleep or is it to keep to the clock times & then work out which one is "more" important, it may not be more important 100% of the time too, but at that time it's about what is better.
Parenting is a balancing act & getting your young baby to sleep is too, so it's a matter of, not so much where to start, but where you think you can work the hardest & get the most benefit for the family unit.
I didn't know BW until my 1st was a toddler & with my second and an almost 2yo around I couldn't do long periods of Sh/pat to settle DS2, but what I did learn was babies really work on predictable routines. I did a lot of the calming down of my DS2 by sitting with my DS1 with DS2 on my shoulder & read DS1 a story, I'd then take DS2 into his bed (or stroller or the car - DS2 did lots of catnaps because of DS1's needs) & that story (mostly) signified time for sleep. I actually never did sh/Pat - doing Shsh did my head in & patting on the back was too hard, so what I did was stroke the top of his head & hummed the same tune. The idea is always to be less involved over time.
My DS2 (who is 12 in 11 days - how did that happen?) strained a ligament in his ankle, it's strapped up & he's doing physio, but the idea is that over time it will get stronger & he needs less strapping, he's about to go back to sport (he does a lot) but will play with it strapped (hopefully to lesser degrees) for at least a month even if it is pain free... sleep with little ones is a lot like that, initially they need lots & lots of support, but over time you test to see if you've reached a point you can lessen the support, sometimes you can't sometimes you can & sometimes (like with teething) you take a few steps back & feel like a lot of the hard work was wasted, but you do see improvement with effort.
Ideally we would like her to be sleeping in her crib for all naps and bedtime. Easier said than done at the moment!
Part of the reason it feels hard is because babies resist change & take a while to learn new routines & the only way for a baby can communicate confusion is to cry & throw in being tired (& at such young ages they go from tired to overtired in 5mins) they cry loads. Hard as it is work with it, created the predictable routine and be relaxed yourself... the more relaxed you are (yep super hard) the more relaxed your baby becomes. If you have a predictable routine before you put your baby down, it doesn't mean early on they will do it without crying, but over time they will improve.
Finally, "baby steps" think how long it took for your baby to be able to control enough muscles to smile & think, if it takes about 6weeks to do that, then it is realistic to need help (lesser and greater degree depending on temperament) to relax enough to get to sleep for their first year of life.