Honestly I don't know what you mean by strong connection puzzles. Based on your text
I am currently developing a puzzle game and I have played a lot of games that have some puzzle elements in them. I found that almost every puzzle game has some "weak connection" type of puzzles. "Weak connection" means that there isn't strong enough evident to support the connection between the solution and the puzzle itself.
This example is also a type of "guess-what-I-am-thinking" puzzle. We actually cannot deduce that the fourth one must be 4 and there are many possibilities, for instance, it could be 1,2,3,5 because it's a Fibonacci sequence or it could be 1,2,3,6 because they're the factors of the number 6. So we have to guess what the developer was thinking through trial & error. Usually, it should be obvious, but it doesn't make sense to me and I think it's a bad design.
I can't think of a single strong connection puzzle. Because the player always needs to figure out the solution that the game designer implemented in the game. And from your description it seems you want there to be evidence of the right solution before actually solving the puzzle. If that is the case, can you even call this a puzzle still? Isn't the whole point of puzzles to figure out the underlying (and therefore most likely intended) pattern? I also wouldn't say that your taxi driver example and finding-the-right-code are of the same category. If the code is found in the same house (or "area") as the lock or otherwise connected to the lock (same person who owns the lock also has the code), there might be no evidence to link the code to the lock. But there is a good reason to suspect that the code is linked to the lock. I can't see how that would be bad design in itself.
But then again: I'm not a puzzle designer and not so much a puzzle player.