As many people above have said, from the DM's side of the screen it really is the defenders, controllers and leaders who mess with your plans the most, and it really helps to let the players know that whenever possible.
I think this is an over exaggeration. Defenders and controllers mess up the DM's plans for a few rounds. Leaders change the momentum of battle back towards the PCs. But, Strikers change the course of battle permanently more often than any other class. As DM, nothing messes up my plans as much as NPCs dying quickly. Partially or fulling locking down a foe is nothing compared to killing it.
Although every role is important, people rarely cheer for the left tackle. They cheer for the quarterback. Even with the knowledge that the left tackle helped set up the play.
When the controller knocks down a foe, but the foe gets back up and charges a PC anyway, it doesn't have the impact of the striker blooding or killing a foe. Or when the controller slows a foe or dazes a foe and the foe still gets an attack.
When the defender takes a hit from a marked foe, the players often do not know if the foe attacked the defender because it had to, or because it would have attacked the defender anyway.
Every player adds to the overall win and every PC does get opportunities to shine, but the encounter in / encounter out PCs that tend to get the decisive portion of the take downs are the strikers. It's the second most necessary role because damage is 99% of the time (shy of rare NPC negotiation or Intimidate if your PCs do that), the only way to win a battle (even running away is usually based on damage). Leader is the most necessary role because it's the one thing that can overcome NPC focused fire, but striker comes in a close second.
Note: To verify this, go play Lair Assault. Balanced teams tend to get crushed because they don't have enough healing/damage mitigation and they don't do enough damage (another form of damage mitigation). Teams consisting mostly of Leaders and Strikers breeze through it.
If NPCs had true strikers (i.e. monsters that did 1.5 to 3 times the damage of other monsters) like PCs do, PCs would lose a lot more than they do now. Brutes and lurkers typically do 1.2 to 1.33 times the damage of other monsters.
Players are so used to the high damage output of strikers that they do not even realize how significant to winning encounters faster it truly is. It becomes commonplace to some people.
A low level party with 2 defenders, a leader, and a controller will typically take out a same level group of foes in 5 rounds, sometimes 4 if the controller can target multiple foes. A party with 3 strikers and a leader will typically do it in 2/3rds of the time and take less damage doing it. There's a major difference in dropping a third of the rounds off an encounter, and the defender having an AC 3 better. One is a 33% decrease in damage from the NPCs, the other is a 30% decrease in damage (50% chance to hit drops to 35% chance to hit) for 2 or 3 foes (with 2 defenders, instead of all of the foes). A party with 1 defender, 1 leader, and 2 controllers will often get crushed because the controllers will be targeted frequently and fall frequently, using up more leader resources.
Strikers not only kill foes faster and end the encounter faster, they are one of the biggest damage mitigators (even better than defenders) because they end encounters quicker, hence, the foes have fewer attacks against the PCs and do less damage to the PCs.
A party without one or more strikers will win the easy fights, but they will drag out. The hard fights will seriously impact such a party because even with focused fire, it typically takes 3 or more successful focused fire hits (sometimes a lot more than 3 at high level) to take out each foe. And this becomes more and more obvious as PCs increase in levels.
Football has a similar analogy. A team with a mediocre tackle and a great quarterback will win more games than a team with a great tackle and a mediocre quarterback. All positions are necessary and all positions have opportunities to shine, but some positions help win more than others.
Note: There are exceptions to this general rule. There are players who play tactically brilliant and optimize their non-striker PCs and significantly impact an encounter in many ways. But as a general rule, Strikers impact encounters more than most other roles most of the time. And the reason for this is that Strikers tend to do 1.5 (at lower levels) to 3 times (at higher levels) the damage of other roles (shy of a few nova rounds that some other roles can sometimes manage). If Strikers were more like Brutes and Lurkers at 1.2 to 1.33 times the damage of their peers, then it would be a bit more balanced. Course, that would require a slight adjustment to monster hit points, etc.
