It might be worth discussing what constitutes an advantage now, since it is a big issue with the OP. A good definition for advantage is "allowing play to continue when an offense occurs, and the team that has been fouled will be afforded an attacking benefit from the application of the advantage [my emphasis]." The question to ask yourself is this: which option benefits the fouled team more: stopping play and awarding a free kick; or allowing the team to keep clear and effective possession?
Did you catch it? The key words here are "clear and effective." Possession must be clear and it must be an effective attacking opportunity. That's why advantage in the middle of the park is so rare and that's why it's even rarer when the quality of the players is so low. There is no reason in the world barring a full-on break-away opportunity to play advantage for a team of 12-year-olds when they are still 40-60 yards away from goal. None. They can't kick far enough, as a general rule, to make it an effective attacking opportunity. Far better to stop play and let them play a set piece.
IFAB has identified 4 key criteria when determining whether to play advantage. Here they are:
Severity of the Offense: In general, the more severe the offense, the clearer the advantage must be. A small, trifling foul? Then the chances of playing on are greater. A two-footed, studs up, locked-leg tackle to the inner thigh of the defender? The ball had better be on the goal line between the posts with nobody around it for you to play advantage. The gradient continues in between. In the case of OP, there is some discussion as to whether the tackle constituted a careless or reckless tackle and it's not clear from his description of the event (we're missing the point of contact, speed, and force). In this case, I can't make a determination about whether the OP's situation was one which maybe should have been stopped right away due to severity or if playing advantage was fine.
Proximity to the Goal: Where on the pitch did the offense occur? Nearer to goal, more likely to be an advantage. Again, in the OP, we see that the incident occurred near the halfway line which is quite far from goal. As mentioned above, this is especially far for the age group who mostly have not developed enough physical strength to move the ball such a great distance quickly.
The Chances of an Immediate Promising Attack: What are the chances that the attacking team immediately springs an attack? Well, look at a few indicators: the number of teammates in the area, the number of defenders, the amount of open space, and, of course, the quality of the players. When I read the OP, I can see how there might have been an opportunity for an immediate promising attack, assuming that the winger was isolated one-on-one with a defender and there were numbers driving forward for the attacking team. This is probably the only criteria that the OP meets for playing advantage.
The Atmosphere of the Match: When the temperature of the match rises, the use of advantage goes down. Instead, we want to slow things up when things get too heated. In the case of OP, knowing who the player that was tackled was could be instructive. The player who was tackled was a known instigator, and the referee knew that this tackle was a sort of revenge tackle because the player in question had left a bit in on other tackles. This needs to be recognized for the danger point that it is, and that raises the severity of the incident to requiring a stoppage. In other words, this criteria was not met.
In order to successfully play advantage, the referee should be satisfied that all four of these criteria are met. In the case of OP, the maximum amount of criteria met are two of the four and it is likely fewer than that. In fact, there's a real possibility that none of the criteria are met. The point here is that there should never have been an advantage in this case due to all of the above and that this is a good, instructive example of how "letting the game flow" can be a powerful tool, but, like most powerful tools, extremely dangerous when not used correctly.
All of this is not to say that the referee in the OP had a 'mare or is a terrible ref. We all get things wrong from time to time and it is entirely possible that he has never been taught the criteria and the correct method of using considerations to make decisions. Hopefully you will all find this helpful and instructive.