Yes the electron is coupled to the ambient by EM. Each electron 'notices' what all the surrounding electrons are doing. It's how we communicate with Mars. Electrons are moved around in the transmitting antenna and every other electron in the universe 'notices'.
Thermal radiation is EM at some point along with conduction, convection.
In the model, the noise source is shown as something beginning or arriving internally in the resistor rather than an externally driven substance that the electron is coupled to. The model's series voltage source or parallel current source is not labelled as 'externally, the ambient' that the electron is coupled to. The source is labeled as a structure internal to the resistor.
In the electron's stable bound state, if the underlying electron model were always loosing energy to radiation from movement of the electron in its stable bound state, (say H, one proton with one electron), this would imply the instability of H, or proton destruction. It would imply the bound state is not stable over the long term as it looses energy in the aggregate and changes form, ultimately to heat.
If this were so it would have happened already and the universe would be a no stable bound state gas cloud, having given up its electron binding energy to heat as radiation to the ambient. This is not what is observed. The stability over the long term and non radiation of H is observed.