The globalization of the Integrated Circuit (IC) supply chain, driven by time-to-market and cost considerations, has made ICs vulnerable to hardware Trojans (HTs). Against this threat, a promising approach is to use Machine Learning (ML)-based side-channel analysis, which has the advantage of being a non-intrusive method, along with efficiently detecting HTs under golden chip-free settings. In this paper, we question the trustworthiness of ML-based HT detection via side-channel analysis. We introduce a HT obfuscation (HTO) approach to allow HTs to bypass this detection method. Rather than theoretically misleading the model by simulated adversarial traces, a key aspect of our approach is the design and implementation of adversarial noise as part of the circuitry, alongside the HT. We detail HTO methodologies for ASICs and FPGAs, and evaluate our approach using TrustHub benchmark. Interestingly, we found that HTO can be implemented with only a single transistor for ASIC designs to generate adversarial power traces that can fool the defense with 100% efficiency. We also efficiently implemented our approach on a Spartan 6 Xilinx FPGA using 2 different variants: (i) DSP slices-based, and (ii) ring-oscillator-based design. Additionally, we assess the efficiency of countermeasures like spectral domain analysis, and we show that an adaptive attacker can still design evasive HTOs by constraining the design with a spectral noise budget. In addition, while adversarial training (AT) offers higher protection against evasive HTs, AT models suffer from a considerable utility loss, potentially rendering them unsuitable for such security application. We believe this research represents a significant step in understanding and exploiting ML vulnerabilities in a hardware security context, and we make all resources and designs openly available online: https://dev.d18uu4lqwhbmka.amplifyapp.com