For sure there are some great 14.1 players to teach the game as you said- to get the balls open as early as possible - that style works real well when the break shot leaves only minor clusters left with 2 or 3 balls to break out. However, there are also a lot of very good 14.1 players who prefer not to execute secondary break shots until they have cleared off a good portion of the loose balls- so as to avoid creating new problems when the broken clusters mix with the already loose balls.
This week we saw a lot of the latter style where theses players were passing up early secondary breakouts until some loose balls were cleared off the table- they were also passing a lot on secondary break shots from below the stack - preferring to break out those clusters when they got the cue ball to the middle of the table. This was commented on several times this week.
Of course, opportunity and layout may dictate earlier breakouts of secondary clusters, but i think keeping balls clear of EACH OTHER and keeping pocket paths clear are the primary goals with the first 9 or 10 balls in the rack- then the priority is aligning a simple run out path to the next primary break ball - assuming you were diligent enough to Leave the correct balls on the table.
A players strategy depends on many things. The conditions of the table and balls, his ability, if he's in stroke etc...A pros strategy will differ significantly from a beginner.
As a general rule, I think the aggressive and somewhat "brute force" strategy is more tolerant of player error, meaning it will allow more balls to be run by a middling or low level player. At the highest level this is no longer true, and more caution may be adviced, as small errors are much more severely punished. For two C players, a player running 20 and then missing is not a catastrophe, but is rather a good start to a game. For two pros the same run could qualify as a disaster, if he leaves any sort of shot to the opponent.
At the highest level, the thing that is likely to stop a run is a "bad roll", usually taking a chance on going into something, or accidentally doing so and ending up out of position. Picking around the balls, rather than going into them, biding ones time etc. is a way to lower the risk of such mishaps.
For lower levels of players, run stoppers, are more likely to be misses of some sort, usually provoked by having to difficult a shot (tough shot or combination of tough shot and tough position). What qualifies as "tough" is also wildly different for the pro and amateur.
My contention is merely that by trying ones best to get the balls open, one is more likely to have more than one shot, and be less severely penalized for inaccuracy in position. Trying to avoid mistakes for a low level amateur is hard, because the player may not have the knowledge to foresee possible bad outcomes, or may not be able to hit the balls with the correct speed, even if he does see such things. His best hope (IMO) is to get the balls open and stay within his abilities while picking the balls off.
Going into the balls hard, spreading them open allows the player to build his confidence, to hit the balls in a positive fashion and to reduce "paralysis by analysis" which often strikes the beginning students of the game. Even if you are not a good player, I think running balls and having an aggressive mindset is more likely to make you a winner. At lower levels, you won't play perfect safety anyways, and 20, 30 and even the occational 40 ball run will add up very quickly to a solid victory.
Just my opinion, as always.