It's not that you absolutely need snipers, with enough smoke and ammo spraying you can get through the game,
Ammo spray? When I got lasers my Assaults pretty much couldn't miss.
You may also want to check your soldier stats, a sniper with 22 accuracy and 21 weapon skill is a total putz who can be out-shot by a grunt with a shotgun. Your good snipers will start with a 27+ accuracy and a 26+ skill
I recently changed what I go after in a soldier I started going for physical stats over skills and my first team is composed of the people with the highest speed and accuracy combination. My last sniper I had preformed well and was worth having on the team till I got plasma weapons and everyone else could out snipe him easily then I got the EM rifle, but it was far too late and the sniper had fallen so far behind that he wasn't worth using anymore because the missions were getting too dangerous to has a soldier that was all but useless most of the time.