There are separate issues at work here:
0) The individual LEDs of WS28x LED strips are enabled by an on-board control circuit. This way the LEDs themselves are technically not directly connected to the GPIO pins. The current limit of GPIO pins is therefore not a problem. Power and data lines are separate.
1) The LED strip - to be technically correct - is therefore not powered but controlled by the GPIO pins. However if the LED strip is connected to the power pins on the 40 pin GPIO header all the current the LEDs require needs to be routed through the traces on the Pi's. 30 LEDs with 50 mA each amounts to 1.5 A in total. That should be in the realm of the possible. The max amperage of header pins is listed with 1 through 3 A (depending on which product you look) and the current rating of the Pi's 5V header pin is about 2.5 A, see Raspberry Pi Power Limitations with the caveat that the Pi Zero got no Poly-fuse; still I would argue that it is safe to assume a current carrying capacity in the same range anyways.
See also: Powering WS2812B LED strip from Pi brownouts
the LEDs are susceptible to voltage drop caused by changing CPU usage.
I would not expect that to happen, unless the power supply itself cannot provide the current. According to Raspberry Pi 3 vs Pi 2 power consumption and heat dissipation we would expect a change from 0.1 A (idle) to 0.25 A (under load), add another 70 mA if WiFi changes from inactive to active, see Power consumption of Pi Zero W?. So the change is quite small compared to the LEDs' current.
Best bet is to check the power supply's capacity and the wiring from the supply to the Pi (many questions here deal with low quality and low cross section USB cables).
2) It is noteworthy that the WS28x have 5 V logic levels. This WS2812B datasheet list 0.7 * VDD as minimum high level. With VDD = 5 V that's 3.5 V which is higher than the Pi's maximum high output level of 3.3 V (let alone its specified minimum high output level). The use of a 3.3V to 5V logic level converter is highly adviseable.
See also: Powering up a WS2812 led strip with different power supplies