Can solving lots of mate in one (or two) problems be beneficial? I am a ~1500 player on chess.com (so I'm in the online beginner-intermediate range).

There is a book titled "Chess: 5334 Problems, Combinations and Games" by Lazlo Polgar, father of Judit, Susan and Sofia. From amazon:


  Chess analyzes more than 5,000 unique instructional situations, many taken from real matches, including 306 problems for checkmate in one move, 3,412 mates in two moves, 744 mates in three moves, 600 miniature games, 144 simple endgames, and 128 tournament game combinations.


Mate in one and two problems aren't usually too difficult to solve with experience. But my question is, does the difficulty of the problem matter? Isn't pattern recognition a crucial part of studying chess? I feel like I can get a lot more from simple mate in one (or two) problems then just solving them, and going to the next problem right away.

For example, these simple situations seem to be really beneficial, if I think about them as positions I want to achieve in real games, and if I try to analyze the position.

Or maybe I am dead wrong about this, and I should concentrate on doing problems, that are difficult to solve.

Any opinions?
There's a wide array of opinions as to the debate on whether simple problems done repetitively, or more challenging problems are better for improvement at tactics. 

My personal opinion is that both have their place and both improve different essential things. Getting those simple problems down cold is very good for pattern recognition and board vision, but the more challenging problems improve tactical analysis abilities, problem solving skills, and help prepare a player for tournament situations.

As for whether your collection of thousands of mate in 1 or 2's would be very helpful, I'd say that mating patterns are certainly important to practice but I don't know if you really need to train with quite so many. It certainly wouldn't hurt, but you might have faster growth practicing with the mating problems on chesstempo. They have them tagged and you can create your own sets of problems if you're a member.