Sure, they just need to involve some risk and the game needs to have counters available.

In CS:GO an AWP is significantly more expensive than AK/M4 and you risk losing it to the enemy team, giving them a major economical advantage for the upcoming rounds. You have to make it count.

You also move very slowly with it and any movement completely ruins the accuracy. On top of that, there's lots of downtime after each shot so missing is punished heavily.

The AWP can also be countered with smoke grenades. You can throw them from safe locations to cover the best sniping angles and allow your team to close the range with relatively low risk.

Because of the downtime between shots, overwhelming the AWP player is also common. He might get one of you, but at least you can get a good trade out of it.

It's a very powerful weapon and many teams have dedicated "AWPers", but it's still not an overpowered weapon or role, because of the associated risk and available counter measures. Not having the snipers would really reduce the depth of the game.

In games like Battlefield, snipers could easily be countered by aerial support and heavy vehicles. The only problem there is that the game doesn't encourage any actual communication or teamwork and instead focuses on personal goals of each individual player. This creates a balance issue simply because the players can't utilize the asymmetric balance and instead it all boils down to "assault rifle vs. sniper rifle".

The Project Reality mod, however, demonstrated that when people focus on winning as a team, rather than their own personal goals, the balance is there. My squad got sniped from a building, so we simply went behind a wall, asked the chopper to come over and waited. Sure two guys died without much chance to fight back, but it created some very interesting gameplay and didn't feel imbalanced.