It's anything but a false narrative.
First off, I specified forwards, which narrows the equation a fair bit (although a separate conversation could be had about Fehervary being the general exception that proves the rule).
Even you have to use quotations around "young" to justify your perspective, knowing full well that guys like Strome, NAK, and Milano are not what people are referring to when it comes to developing young forwards under Laviolette. Strome has 310 games of NHL experience at this point, and all three of them at least on their third NHL franchise (with NAK on his 4th). Both NAK and Milano will be 27 in May.
Malenstyn is 24,
6 years removed from his NHL draft, and has a whopping 20 games of NHL experience at this point. Claiming him as a Laviolette development success is pretty ridiculous. I won't even get into the debate about whether or not his less than 44 minutes of ice time in the NHL this year was good.
Laviolette's system is not something overly fancy or "complex." It's a pretty standard 2-1-2 spread forecheck (although he calls it a 2-3). You can hear him
break down most of his system yourself if you're interested. There's not really anything in his system that an NHL prospect would not have experienced in competitive hockey before. He's not doing any Adam Oates-ian micromanagement of players, or a novel neutral zone trap like Guy Boucher, or even an old school defensive zone scheme from hockey antiquity like Bruce Boudreau. It's carry the puck in, pass if you can't, dump as the last option, with a 2-1-2 in pursuit, 1-2-2 neutral zone trap in defensive transition, and Box+1 in your own zone. Nothing atypical.
Protas is as close to a development success of a forward the Capitals have had with Laviolette, and at this point he only has 16 career points in 70 NHL games receiving 4th line minutes, and he's actually seeing less TOI (and less PP time) in his 2nd NHL season than he saw in his 1st. He's not exactly going to be running away with a bunch of Calder votes. Turning a D+3 player into a 4th liner isn't exactly beating the development curve. Compare the development of McMichael and Lapierre to similarly drafted Capitals forwards in the past, and how far they're lagging behind guys like Burakovsky and Johansson in NHL experience at the same ages.
You seek justifications for why younger players didn't make it after the fact. "If Lavi didn't play them, they must not have been ready to play the system." McMichael has been debated to death on this forum, so I don't want to rehash that, but suffice to say that his ice time last season had no linkage to how well he was or wasn't playing (and far inferior veteran skaters would often get ice time over him when in direct competition for minutes even after those veterans committed egregious mistakes). Now you're heaping that same post-hoc justification logic onto other players. Snively was praised for "earning his spot" and "seizing the opportunity" last season, but after less than 60 minutes of ice time this year he didn't "earn a spot" and hasn't played a game in 6 weeks.
Your logic is that "rookies can play for Lavi" but if they don't, it must be the players fault. You don't see how that's a self-justifying prophecy? If they play, it's because of Lavi. But if they don't, in your eyes it's because of the player. Yet, we've seen repeated examples (including before his time in Washington) that Laviolette applies different standards to veteran forwards than he does to prospects and younger players. Just look at Kevin Fiala's breakout as soon as he got to Minnesota, compared to what he did under Laviolette in Nashville.