If you are trying to find the current through the 1 ohm resistor i.e. that is your target aim then you can't "re-use it" when converting the current source to a voltage source because it loses its connectivity and that invalidates you being able to analyse the result. The 1 ohm resistor must stay precisely where it was originally or all bets are off.
Using thevenins or norton theorem, we can easily find current through the resistor as 1 amps.
It's as plain as day following night that the current through the resistor is 1 amp without using either theorum. There is 1 volt across it and that means 1 amp. The extra current from the current source just flows into the voltage source.
The equivalent circuit is only equivalent when you are looking at current/voltage relationship between the two terminals where you made the conversion. The currents and voltages inside the converted circuit are not necessarily the same as in the original circuit. If you care about the current through the resistor then you can't do any transformations where the resistor is part of the converted circuit.