Models which try to describe how the AC components of the voltages and currents are related with each other are not images of physical structures. Your problematic T-small signal model works approximately like a fet when only weak enough AC components are taken into the account.

Ig=0 because the voltage dependent current source makes just enough current to compensate the current that the resistor would sink from the gate in case the current source was removed.

Small signal models assume certain proper DC operating point (=idle DC components of the voltages and currents) which are left out in small signal AC analysis. The trick to get those DC components right is called "biasing".

Models which try to describe how the AC components of the voltages and currents are related with each other are not images of physical structures. Your problematic T-small signal model works approximately like a fet when only weak enough AC components are taken into the account.

Ig=0 because the voltage dependent current source makes just enough current to compensate the current that the resistor would sink from the gate in case the current source was removed. The apparent G-S resistance seen from the gate becomes infinite, no matter there's a finite resistor.

Small signal models assume certain proper DC operating point (=idle DC components of the voltages and currents) which are left out in small signal AC analysis. The trick to get those DC components right is called "biasing".