Think Fast, Move Slow

Think Fast
You don't want to over analyze.
You can't afford to think too long when the clock is running.

So, to play better chess, you need to learn to think fast.
That's why you're supposed to be doing puzzles - to improve your pattern recognition. So you can find tactics 'effortlessly' and quickly. If you're doing puzzles and taking all the time in the world to solve them, you're missing a big part of the picture.

Puzzles help you learn patterns.
AND they can help you condition yourself to find those patterns quickly.

Granted, half the battle is finding the answers at all. If you can't find that 2 (or 1) move mate after thirty minutes, then YES, you need to get better at finding the answers first. Speed is not the critical issue.

BUT when you get to the point where you can find the answers, you want to expedite your mental processes for finding that move. You want to find them faster and faster. There are two things to gain from puzzles - recognition of the pattern (the solution) AND decreasing the time it takes to find it.

Move Slow
Bullet games are terrible for your improvement. They're like a 300 calorie drink - tastes great, but it's terrible for you. You learn to premove as many moves as possible. You lament that your quick twitch reflexes aren't fast enough. You get flagged more often than mated. You think the problem is you move too slow...and for Bullet (or Ultra bullet) you probably do. But for any realy game of chess, it's not your mouse speed that needs improvement, it's your thinking speed.

So, move slow!