hah?
More nonsence of puzzle rating
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The 2nd example I can understand. Almost any player will take 1.d6+ as a free tempo move while it is a serious blunder. After ..Kh8 2.Rc3 (still relatively the best) Qe4! 3.Bb1 does not work because of the e4-a8 diagonal opened by the first move. Compare to the same sequence after 1. Rc3! It's the sort of thing even overlooked by GMs because the initial check opening the other diagonal to the bishop is so attractive!
The first example is less clear. It's not that 2.Nf7 looks bad but that the check 2.Qf7+ looks more inviting. After 2..Ne7 both 3.Nf6+ and 3.Nxe7 (and even 3.Ne6) promise a big advantage - it seems. There is solving psychology in there. With several promising moves after ..Ne7, probably one will work. Which one I will see once we get there. Lazy but logical. It comes from a long experience of misguessing engine responses in puzzles. So often the engine does not defend optimally (humanwise) that we hate to waste our time on calculating all our critical variations which are only met with contempt by the engine!
Yes true well i told you like you can solve it without hints IF you are 2000+ like the 2nd example.
Qf7+ is probably the first thing you will see but it is not the answer. Engines seem like they sometimes dont know what they are doing. Sometimes they drop a random brilliant move for no reason that is literally a forced move and they put it as "Brillaint" just becuase it keeps the position stable but in reality it is the only moves that humanly makes sence. Also it can also depend on the depth i saw a puzzle which said winning for white but in reality black can force a draw and the problem here is engines will try to keep moving as long as they can and they don't understand the 50-move rule. But when i increased the depth it DID appear as a draw so it can also depend on the depth sometimes.
IN SHORT: Engines are random and just another creation not even its creators can understand AKA Humans.
The engines are not doing anything wrong in the 2 examples you provided. They gave the right answers and evaluated all the lines correctly. And so does StockFish when you set it to work. The errors are more with the humans causing the ratings to soar to 4000. I tried to explain the reasons it might have happened. Note that puzzle solving humans are commonly under time duress and then tend to follow familiar paths. Even when solving in modules like "Puzzle Rush Survival" where speed doesn't matter, strong players like Hikaru will rush for the training. Quantity has more value to them than quality and they accept some stumbling into traps they would find in an OTB game. Which makes the puzzles seem much harder then they actually are.
I agree engines have many flaws - which are all programmed into them by human handlers. Notably, they shouldn't be fully trusted in selecting good puzzles and deciding on engine responses. These are often senseless to humans and do not help them to become better tacticians.
Issues like 50M are complicated. In games there is a 50M rule, but in endgame problems there is not. If you use the same engines with the same settings for both you may get into trouble. Nowadays there is also a 75M mandatory draw rule in the FIDE laws which adds further confusion to the picture.
https://www.chess.com/puzzles/problem/1249541 Puzzle 
respect
This puzzle is also very nonsence. Like... Just look at it...
These are puzzles i found by filtering with 4000 rated puzzles. I mean how do this puzzle ratings even exist and why are they even 4000? Some of them is just like very easy and 4000 for no reason and some of them are like insane but have same rating... Here are the links: https://www.chess.com/puzzles/problem/1122180 This puzzle is like... idk 1000 and it is 4000?? it is easy I am sure you can do it without hints. While there is this absolutely insane 4000 puzzle which even if u 2000+ pretty sure you need hints. Here is link: https://www.chess.com/puzzles/problem/1249541 Like what the hell is this compared to the other one?? Anyways... If you have more of this ridicuous puzzles and ratings share them here!