Microcontroller driven N-Channel MOSFET voltage behavior

Place the load between the 7.5 Volt supply and the MOSFET drain, and you will get switch-like behavior.

^{simulate this circuit – Schematic created using CircuitLab}

In your current layout, the source is floating depending on current through the LED and resistor. Hence, Vgs is not 0 to 5 Volts as you assumed, but much lower, depending on where Source is floated to at a point in time.

What is needed is for the "switch" to conduct when given a positive signal from the Arduino, thus pulling its Drain node to ground (or close to it), thereby expressing the desired ~7 Volts across the LED + R3.