Limiting the inrush current
Putting inrush protection on a transformer primary is a pretty standard way of doing things because it also handles primary magnetization current inrush problems too.

Avoiding an MCU means having a small timer circuit that, when timed out (having been initially triggered by the instance of power being applied), activates the relay as you show in your diagram.

Or, you could avoid all the complication and use a negative temperature coefficient varistor/thermistor in series with your primary winding. This is a pretty standard solution in many power supplies.