Triggerless $2/1$ neoclassical tearing modes (NTMs), i.e. $2/1$ NTMs that originate from unstable safety factor profiles (with positive classical stability index at zero island width, i.e. $\Delta'_0>0$) and saturate neoclassically under the effect of perturbed bootstrap current, have been observed reproducibly in TCV discharges with strong near-axis electron cyclotron current drive (ECCD). An unexpected density dependence of the onset of these NTMs is newly observed based on the statistics of many TCV discharges, suggesting a certain range of density within which the NTMs can occur. The range is found to broaden with increasing near-axis ECCD power. Based on a different set of experiments and corresponding interpretative simulations with the Modified Rutherford Equation (MRE), a simple analytical model for $\Delta'_0$ has been developed. This model proves to explain well the observed density dependence of mode onset, resulting from the density dependence of the stability of ohmic plasmas and of the ECCD efficiency. Simulations have also quantified various effects and reproduced self-consistently the entire island width evolution of the triggerless NTM, from the onset as a tearing mode at zero island width to the neoclassical saturation as an NTM. The standard terms of the MRE model used in the paper are relevant for NTMs that are seeded by other mechanisms.