This simply isn’t true. Initial conditions for time series models need to be provided as part of the problem definition. They cannot be generated by the system itself, otherwise they wouldn’t be the initial condition.
It’s important to be clear between an ETS model and an AR model. As written, you just have an AR(1) with a funny parameterization – the (1 - \rho) factor on the innovation term can be pulled into the mean and variance terms of the r_t distribution, getting you back to the base case of \tau_t = \rho \tau_t + \eta_t.
An ETS model would instead be written:
Where \ell_t is a latent state describing the level of the system at time t, and \epsilon_t are innovations. In this case you also need to describe the initial level of the system \ell_0, which is a distinct object from the innovations.
You mentioned convergence, but ETS models don’t converge to a steady state in general. This simple model will produce constant forecasts of either the sample mean of the data (when \rho = 0) or the last observed data point (when \rho = 1). An AR model will converge to \tau_{ss} = \frac{\alpha}{1 - \rho}, where \alpha is a deterministic drift component (I guess zero in your case, but maybe not). One can absolutely use that formula to initialize stationary processes – it’s what I recommend for stationary state space models in the statespace module.
Here is an example notebook on SARIMA models, and another on ETS models that might be helpful? I strongly recommend to run the ETS notebook in a colab to playaround with the interactive plots.