The only reason I suggested fixed gear rather than single speed freewheel was without a coaster brake there is no rear brake but a fixed gear bike pretty much has speed determined by the rider's cadence and has a braking ability by reducing the cadence. Relying completely on the front brake is not particularly safe and if you are investing in a rear brake caliper, cables and brake lever, you might as well cough up for a coaster brake rear wheel anyway and get it back to original spec.

Although saying that here in Europe a coaster brake rear wheel isn't too expensive.

https://www.taylor-wheels.com/bike-w...9-black-silver

https://www.taylor-wheels.com/bike-w...-406-19-silver

In fact a 3 speed conversion with coaster brake isn't very dear and this is decent Shimano stuff.

https://www.taylor-wheels.com/bike-w...r-black-silver

3 speed conversion would be tempting for me if that is possible (suitable hub width etc) especially as that pack includes everything including the shifter. Cheap as chips.