Tilting the rotor would not work!
The cyclic (and you could replace the swashplate with electric motors, but it would still be cyclic controls, just less reliable) works by shifting the centre of lift over the area covered by the rotor, and can shift it by significant fraction of the blade length. This is necessary to compensate for the effect of forward flight, which shifts the centre of lift significantly towards the advancing blade (which creates pitch up moment).
By tilting the shaft you could not create nearly as large moments as you can with cyclic. The centre of gravity isn't that far from the hinge, so you'd need larger angle than the geometry allows. With cyclic, the large moment is either created during rolling or pitching or for maintaining speed and the fuselage follows, so the tilt isn't large.
You would also get the full consequences of the gyroscopic effect. The helicopter would tilt to one side to accelerate and to the other side to decelerate! That is because the tilt just creates a moment on the rotor, but the rotor is still a gyroscope that still tilts with (a bit less then!) a 90° lag. The swash-plate deals with this simply by mounting the riders corresponding angle ahead of the blades, with the ultimate effect that the rotor is always trying to align itself with the swashplate (minus the effect of speed).
In fact, many helicopters (e.g. the UH-1 Iroquois (Huey) / Bell 204/205) have teetering rotor and in these the fuselage can't apply any moment on the rotor at all, because (as the name suggests) the pair of blades is allowed to freely teeter on the hub, and the fuselage is just freely hanging below it. Here the property that the rotor is aligning itself to the swashplate is crucial for preventing it from striking either the fuselage or the mechanical stops and getting damaged—when the pilot pushes the cyclic, the rotor starts to tilt, but as it aligns itself with the swashplate, the pitching moment will reduce until the fuselage catches up and the swashplate tilts with it some more.
In fact in all helicopters the rotor is free to tilt in response to the aerodynamic forces and the fuselage is mostly just hanging below it. When the blades have offset flapping hinges (fully articulated rotor) or rely on flex (hingeless rotor) some moment is transferred directly to the shaft, but it is always relatively small compared to just pulling the hub to the appropriate side. Beside dealing with the lift asymmetry due to forward speed, this free flapping of the blades is also necessary to deal with turbulence and similar disturbances.
So no, you can't tilt the rotor mechanically by adjusting the angle to the shaft, you have to do it aerodynamically, and the swash plate is the mechanically simplest mean to achieve that. Separate electric motors would just add more points of possible catastrophic failure.