How do you do it? Get your baby to fall ASLEEP?

TNY • Scientist. Wife. Mom to 🐶 (2013), 👶🏻 (05/16), 👶🏻 (06/18), 👼 (03/20), 👶🏻 (12/20). Starting my advocacy for the disability community 💙💛.
Okay, so I have a very stubborn 3 1/2 month old who will not nap and now is resisting bedtime. How DO you actually get your bambino to FALL ASLEEP?
I have tried everything: routine; sleep sack,; minor crying it out; protesting it out; rocking; shushing; holding; bouncing; putting him to sleep awake, drowsy, sleeping; car rides; stroller rides; wraps; crib; sleeping in my bed beside me...  Nothing is working for us. We are even back tracking our sleep successes at night (he would go from 3-7 hours in a row).
As a FTM I have done a few things wrong like allowing him to fall asleep at the boob during the early days but he always just woke himself up anyway. Also I never "forced" naps before so he would be pretty much all day except for short cap naps (at the boob) and have started trying a nap routine the past few days...but it's not working. I know he is seriously overtired.
Basically, he protests for hours OR gets so upset that he screams for hours. I am not comfortable with a full cry-it-out so I pick him up after a set time and it can take him an hour to settle back down in my arms. The only thing he wants is the boob but even that doesn't keep him asleep. He almost falls asleep many times but then wakes himself up! 
Help?! I have a great sleep routine (I got some causal tips from a sleep consultant) but can't get him to sleep. How do you literally get your baby to fall asleep not the routine you use... Thanks in advance!