The zerg absolutely require strategy to play. Its of a different kind, though! (Talking about high-level SC1 play here.)
For example, did you know Zerg are the race the most dependent on flankings, envelopings and pre-positioning their army? Also, they are the most mobile of the 3 races, and they are the most adaptable (because larvae can morph into anything, whereas Terrans and Protoss are bottlenecked by the type of production facility they need).
Zerg units are also very fragile. This means that if you don't position your units properly, you can get completely wiped out and do no damage whatsoever. This is why flanking is so important. For example, if you send 50 zerglings at a terran infantry force head on, all of them will die and not kill a single marine, if the marines (with medics) have reached critical mass. However, if you envelop from the rear and the front as soon as the same number marines are on the move and straggling a little, you can wipe out the entire force (you will still lose a lot of zerglings though, that's just the nature of zerg... lol).
Also, flanking the enemy somewhat avoids brutal AoE style attacks like Psi Storm and Siege tanks
Zerg have one of the most intersting spells in Dark Swarm. Dark Swarm is vital because it creates "safe zones" so the Zerg army can actually compete with the concentrated ranged firepower of Terran in the late game and push the Terran army back. A handful of zerglings and just two lurkers under a dark swarm can do serious damage to a large terran infantry force.
Finally the linchpin of master-level Zerg play is timing and larvae management. Zerg have the msot interesting production system because of the larvae. Terran and Protoss produce workers and units in parallel through different production queues, but the Zerg cannot. The advantage the Zerg have is that if they fully devote their larvae to either workers or units, they can produce either one faster than if the queues were parallel. A true Zerg master is distinguished by the ability to know exactly how many units are needed to avoid dying and then pour the rest into workers- or, conversely, to know exactly how many workers are needed to fuel the machine and then pour the rest into units, and most crucially, where the line between these two phases lies.