Oh, sorry to hear you're feeling miserable
I totally understand the frustration and how draining it is to try and extend every nap. Maybe you can focus on one nap at a time so just work on the morning nap? Do you think you have the energy for that?
You've come so far, it has been less than a month since you began ST I think, and although you are patting you also say he is going to sleep on his own, no patting, eventually - this is GREAT! Honestly this is a process of gradually teaching him to have the confidence to sleep alone and gradually weaning the patting - and you are making progress through that process.
Patting is not considered a prop because it is relatively easily weaned. you are doing just the right thing in patting when he needs reassurance and help then reducing and stopping as he falls to sleep or before. I found even with a fully independent sleeper there were times I would pat (adapted version) fully into sleep if he was struggling with something.
I know this feels amazingly hard for you right now, yes there may be a missing piece of the puzzle as you say, and yes it may be a longer first A time, or it could just be down to a very normal developmental phase which makes naps tricky at this age.
I would increase. I know mine had his best naps at 2hr 30 at this age. Remember to increase gradually and that it will take several days for any change to settle down, there could be OT naps initially whilst he adjusts.
The following A times are very short - were these following cues? I wonder if you can keep them longer despite the short naps?