Can anyone suggest some cost-effective circuits? I am thinking of using the Hall effect sensor like the TMCS1107A4U, but this seems a bit expensive for what I'm trying to do, I don't need to measure the current, I just want to detect when it's flowing. Current sense resistors and appropriate amplifiers also seem a bit expensive (relatively).
1mA-5A is a significant range, but you can get low offset opamps and low offset current sense amps.
The cheaper current sense amps run up to 26V common mode, so you would need to externally level shift those, or you can level shift a 1.8-5V opamp.
Offsets below 10uV are easy to find, if you map 1mA to say 4x that or 40uV, that's 40milli Ohms of sense resistance.
At 5A you then drop 200mV for 1w loss.
Opamp examples : lcsc have AD8628ARTZ(UMW) for 22c/50, you need to level shift that.
The offset voltage is usually spec'd at Vcc/2, and it can get worse near the rails. Not all suppliers will define that.
If you want
lowest possible offset, you would float that opamp above and below the rails, and level shift with a PMOS or PNP or optocoupler