Then 5th edition came out and it was immediately obvious that it had that 1st edition feel but with all the bugs worked out. I play it with my kids and it's awesome.
Before feats, the raven can fly 40' a round. Between that and items to allow for empowered and maximized fireball and scorching ray (I think?), our wizard regularly drops between 60 and 120 damage per round.
We basically have an Apache attack helicopter.
As late as 3.5e and Pathfinder, you could easily spend 5 minutes figuring out how to resolve a single 6-second action according to the rules. Say I'm grappling someone (complex rules) and move them across rough terrain (5-foot step or not?) while threatened by an enemy combatant (attacks of opportunity) with while under a Bard buff (pluses to defense)… figuring out who attacks whom when and how can easily involve flipping to many different sections of the rule book. Whereas in 5e half those rules don't exist and the remainder are simplified.
My understanding (not from experience) is that previous editions had even more complex rules than 3.5e, and 4e simplified the rules in the "wrong direction" (away from role-playing).
In first edition the Armor Class went from 10 downwards eg AC -2 was awesome. And you had to look up tables (before 2nd edition added the slightly simpler THAC0). That was changed such that numbers always go up and similarly, higher dice rolls are always better (in some circumstances in 1e you wanted lower numbers).
The concept of advantage or disadvantage. Sometimes you'd have to do a bunch of calculations eg -4 for being invisible, +2 for this, -1 for that, etc. Now, you just figure out whether someone has an advantage or disadvantage. If neither, it's a normal dice roll. If advantage, roll 2 dice and take the highest number. If disadvantage, roll 2 dice and take the lower.
The best bit of advice I can give is to let the kids be creative and don't be too fussed about the rules. It's like improv - any idea is a gift - even if it's an outlandish idea, let them do it, make up some difficulty score and when the dice come up with a failure, try to be as creative and funny as possible in describing the outcome.
Is it generalised/generalisable enough for use with other source material and RPGs? In other words, is it D&D specific?
One tip that I use in every game that is incredibly useful and isn't in this book (or Sly Flourish's other great book, "Dungeon Master Tips") is to have three NPCs who represent "Money, Power, and Fame" and are always on the move. Their storylines develop even if the players are not interacting with them. Remember "Fame, Money, Power" and create a single NPC to represent each of these three things. For example, you could have a local baron who is hungry for power, an evil guildmaster who is trying to expand his influence over local trade routes for money, and a retired, good-hearted adventurer who helps out new adventurers at the local adventurer's guildhall because he enjoys the fame and recognition.
Each of these three NPCs will have some immediately obvious conflicts which will generate storylines you can very easily improvise. I don't even need to give you an example since you can probably think of some conflict in those three examples just off the top of your head. These three NPCs also have easy access to each of those three things (money, power and fame respectively) and typically, at least some of your players will be motivated by this to some degree. That makes for an easy carrot-and-stick for players. Do your players need money for an expensive spell component to save the mayor's daughter? You might find the guildmaster willing to pay a handsome reward for completing a small side quest, for example having the players bluff or sneak their way into the baron's palace during a party and steal some important documents. The local baron of the land can probably give you the power to raise a small militia to stop a rampaging band of orcs in the area. Does the king's court have no idea who you are? It turns out the retired adventurer might be willing to write a note on your behalf, if only you could do him a small favor...
And the important thing is that you should always be moving these three NPCs along their life paths even if the players aren't around to see it. This is very easy to track with your index cards. And if one of these NPCs is killed or retires or somehow drops out of the game, you should come up with another to replace them. Perhaps the evil guildmaster has the baron assassinated, and now that he's had a taste of power after his guild is given control of the barony, he becomes your "Power" NPC, and a rival guildmaster becomes the new "Money" NPC. See how easy that was? And you can do this in your sleep really, so it works out great for the method described in The Lazy DM.
but then again I am so used t dnd having started in the 3 brown books days
I cannot say to which degree the game systems actually helped in practice; I switched groups between editions, and my 4th edition group was 90% improv theatre people. That there would be more roleplaying was pretty inevitable.
I also haven't played 5th edition so I don't know the changes there.