what is the problem of grounding a 1 ohm resistor to a power supply?
The higher the current is the higher the load ground will be above supply ground. This may be acceptable in isolated systems, but in systems that connect with other devices there can be a real problem if current goes above a few hundred milliamps.

Also, the higher the current is the lower the supply voltage will be. This means that your system will brownout itself easily if it draws too much current.

The way around both of these is to use a much lower resistance, 50mohm or less, and an in amp shunt amplifier (e.g. INA213) to raise the difference to measurable levels. Integrated solutions (e.g. INA250) also exist.