I think the lower you go, the more specific it gets.As is usually the case, what might be true for most is not true for all. Everyone has weaknesses preventing them from getting to the next level and they can be quite different.
Putting is absolutely relevant for an over 100 golfer who is a very poor putter. In my case, poor lag putting can ruin an otherwise (relatively) decent round of proximity to the hole. As a high capper, 100 can be broken with some poor shots, a penalty or two, and even a blow up hole or two. But there can't be too many of those and the rest of the round has to be a bit better than average.
On the occasions when I've broken 90, rarely did any part of the game stand out as exceptional. There was just enough consistent execution throughout the bag mixed in with a little luck.
That said, my experiences obviously do not apply to others.
To break 100, you certainly need to work on whatever your weakest point is that's keeping you from doing it. If you're giving away a lot of penalty strokes off the tee, you need to work on keeping your drives in play; if you're wasting 5-6 strokes around every green, you need to improve your short game; if you're 3 and 4 putting every hole, you need to work on your putting, etc. You don't need to have an overall great game, but you need to fix whatever part is the most terrible/hurting you the most. You can get away with a lot and still break 100 - if you double bogey 9 holes and bogey 9 holes, you just shot a 99.
When you get down to breaking par, every part of your game needs to be on point. Now it's not just keep your drives in play, it's try to put your drives in a specific place to set up the next shot; it's not just getting somewhere close to the green, it's trying to get on the green and close to the pin; it's not just cutting down on your wasted greenside strokes, it's trying to get up and down any time you miss a green. The margin for error is now a lot smaller and it's going to be hard to break par with any glaring weaknesses in your game. Just keeping the ball in play and keeping it in front of you doesn't cut it anymore.
But you're correct that everyone has their own individual weaknesses that can keep them from hitting that next barrier. It's up to them to diagnose what they are and work on improving them. Take five guys of similar handicaps who are struggling to break whichever score barrier, and they'll probably all have different reasons why they're not breaking it - but as the scores go lower, those differences are a lot smaller and more precise.