Why Counting is Futile
I'm using National Master Dan Heisman's version of counting (for those who are curious). Note that he also points out some of the problems of counting that I note below. He defines counting as looking at the number of attackers and defenders on a piece to know that if an exchange occurs, you can see if you'll remain even, be up in material, or down in material. He stresses that this is important for beginners to understand and implement this process.
For example in the position below (White to move) White may think "If I do nothing, he has two pieces (bishop on c5 and knight on c6) attacking my knight, while I only have 1 defender (Queen on d1). So I need to move my knight off the d4 square or give it a second defender."
Now you may think "well that's not a problem, as a new player White came up with a reasonable plan to deal with the threat at hand." On the surface this seems fine, however it creates a trap that may hamper long-run improvement, cause them to miss moves, and occasionally outright lose games. What do I mean? Well, players get so used to counting that they don't stop to actually calculate.
For example (white to move) below White may think along the lines of "The counter" that he wins material by taking the pawn on c4 as it is only defended once but is attacked twice. Sometime during the sequence though he will notice that his knight is actually pinned and he lost his bishop for a pawn, or may even walk into checkmate.
If he had actually been a calculator who visualizes where the pieces go in the case of an exchange, (with experience/practice) he would be able to visualize that 2 moves in if his knight leaves that diagonal his king would be in danger. While this is a simple example and one could argue the counter could just "remember that his knight is pinned," the fact is there are often many in between moves, attacking moves, and other complications that can arise over the board and a player can easily forget. At the end of the day to improve your chess you need to actually calculate and visualize until it becomes automatic.
Solve the puzzles below that a mere counter might (or might not) find unintuitive.