No, not that type of breaking. That type! So...in your current or past training, did you or have you broken stuff? If so, * What sort of breaking did your style engage in? Not all breaking is the same. There's a huge distinction between "breaking" in the sense of breaking a board with a fancy aerial kick, and "breaking" in terms of breaking roofing tiles with a downward palm strike. * Did you enjoy it or find value in it? If so, what did you like about it or find useful? * As that style approached it, how did it tie in with the rest of the training?
Hmm, when I was about 7 or 8 I was enrolled in I believe TKD, though I could be misremembering--it is possible it was a Karate place. (I had 2 martial arts as a young kid: one karate place (no idea which style) and one TKD place.) We did board breaking with punches & kicks. It is hard to remember back (nearly 30 years now), but I'm pretty sure I found it enjoyable--I always did like turning objects into smaller, less organized objects*. As to value in it, I doubt I got any value out of it. I have no recollection of how it tied into the style... * E.g., when I was a teenager I for some reason found it entertaining to take an empty tic-tac container, place it on my palm so the label is facing up or down, and then make a fist with my other hand and punch the tic tac container 3-6 times, stopping when I figured I'd just be impaling my hand on jagged edges... I'd sometimes end up a bit cut up on my knuckles but found it fun anyway.
We break for gradings and in class occasionally. It's fun and a good test of accuracy and power under ideal conditions. The boards we use are rebreakables, plastic boards with a slide together join. They're not easy to break; the easiest ones can take a 12+ stone person standing on one foot directly on the join and they won't break, so you have to hit hard, and if you miss the join by even a small amount it requires much more force. So as a test of accuracy and power (and confidence) it's fun. Nothing to do with self defence as such, other than the attributes it develops, but fun.
In kungfu we did breaking for higher grading. Downward palm strike onto 1or 2breeze blocks Front kick a breeze block And 1or 2roof tiles broken onto your rib area. Building up to it I thought it tied in with what were did, although there wasn't really much prep training before the event. We did plenty of power work on pads/ shields, which transferred to the breaking. There is prep that needs to be done for the blocks before: keep them warm and very dry so they stay brittle. Unfortunately I didn't know this before my black belt test, but I still managed to break 2 blocks on a second attempt, but my hand suffered after It's an interesting tool for focusing and committing to power in a strike that mentally I wasn't able to get into of were to do the same on a bag and try for maximum power. Since doing it I feel I can focus for a power shot better, but I still would need to work on doing it without a delay during that mental focus stage. The only parallels I can draw on other training methods is when working a drill to kick or punch someone with a kick shield to make them move as far as possible.
In Taekwondo, we did board breaks at each color belt level. Each level had a specified technique (like front kick for green belt, side kick for blue, back kick for red, and spinning heel kick at all red levels. For 1st dan and up, we required a multiple station break). It was a great confidence check on key techniques and required good form. It was also fun while being a bit intimidating. When I went to Korea, we did breaks on lots of techniques with the rebreakable boards. In Hapkido in Korea, we broke the ceramic roof tiles in stacks.
The best case for breaking I ever saw was a shy teenage girl that was clearly awkward with being physical and lacking in confidence. She could hit pads and spar but never quite felt or realised she had any power or ability. She broke a re-breakable board and you could see the realisation wash over that she actually had some power in a way that other forms of training didn't quite reach. I'll always value breaking for that kind of ability to make people realise their potential (although obviously it's not the only way). I often feel that people who are pretty good at fighting or have reached a good level lose sight of these kinds of little confidence steps that they would find trivial.
This is exactly how my TKD club did it. I thought it was helpful for checking proper form/precision/power/etc, but like you and Smitfire said, the biggest thing was confidence, especially with more timid people or with kids. It wasn't a primary part of our training or anything--padwork and sparring probably got 100x the training time as breaking did--but it was the last step in reach belt test, typically with a more complicated kick at each rank.