Will a phone or other wireless device increase its EMF output level if it receives a weaker signal from the Eco router, so that there is an unintended increase in radiation? Similar to how a mobile phone transmits at higher power when the range is poor? In other words, don’t the receiving devices have to ‘work harder’ to maintain the connection, potentially increasing EMFs locally?

No, because there is no well-functioning wifi power control in modern phones and other wifi devices. Such a control would have been much better, also for battery life, but apart from the 802.11h control for 5 GHz wifi which is almost never in operation, there is no such control in the wifi standard. Therefore, phones and other wireless devices always ‘bluntly’ transmit at maximum wifi power, regardless of the signal strength received from the router.

You can easily verify this yourself by measuring a modern smartphone or other wifi device with an RF meter. Measurements even show that wifi is the strongest source of electrosmog in modern smartphones. See also 10 tips to reduce electromagnetic radiation from your cell phone and wifi. For some devices with wifi, especially Windows laptops with an Intel wifi adapter, it is possible to reduce the transmitting power in Device Manager.

The signal strength received from the router, is determined by its transmission power setting. For more information, see Does the Eco router reduce the peak power of the pulses?