Knowledge Distillation (KD) is essential for compressing large models, yet relying on pre-trained "teacher" models downloaded from third-party repositories introduces serious security risks--most notably backdoor attacks. Existing KD backdoor methods are typically complex and computationally intensive: they employ surrogate student models and simulated distillation to guarantee transferability, and construct triggers similar to universal adversarial perturbations (UAPs), which being not stealthy in magnitude, inherently exhibit strong adversarial behavior. This work questions whether such complexity is necessary and constructs stealthy "weak" triggers--imperceptible perturbations that have negligible adversarial effect. We propose BackWeak, a simple, surrogate-free attack paradigm. BackWeak shows that a powerful backdoor can be implanted by simply fine-tuning a benign teacher with a weak trigger using a very small learning rate. We demonstrate that this delicate fine-tuning is sufficient to embed a backdoor that reliably transfers to diverse student architectures during a victim's standard distillation process, yielding high attack success rates. Extensive empirical evaluations on multiple datasets, model architectures, and KD methods show that BackWeak is efficient, simpler, and often more stealthy than previous elaborate approaches. This work calls on researchers studying KD backdoor attacks to pay particular attention to the trigger's potential adversarial characteristics.