Arduino RFID reader and pin current sourcing
Firstly, the 330uH inductor you already have is almost certainly not suitable unless it is an air cored inductor wound on some kind of inert magnetic material or an open ended ferrite rod of reasonably low permeability. Check what it says on the circuit you linked to. This coil has to transmit and receive so it needs to be likely a non-bought-in (and likely hand-wound) part.

D9 excessive current in simulation - you need to current limit the top circuit because it's driving a series resonant coil and capacitor and this will act like a short circuit at resonance. Maybe try 100 ohms or a bit less.

In your alternative oscillator circuit you've got Q3 upside down in your simulation of the RFID transmitter - that will cause excessive base currents through Q3 and may easily account for the excessive current from IO D9.

Your tag circuit might also benefit from a 100 ohm in series with the collector of Q1 too.

What does the waveform look like when the tag isn't responding - does it still show amplitude modulation issues?