ESPHome has firmware for the BT proxy:

https://esphome.io/projects/?type=bluetooth

The I installed the xiaomi app on my phone, I think it was called zepp life or something for the scale I have, you do the initial setup of the scale on there, and then it’ll be picked up by the Bluetooth proxy.

Yep, pretty much. Just get an ESP32 and not an 8266.

I have it plugged into an old Nokia charger and hidden in a cupboard to service all the Bluetooth temperature and door sensors.

When I use the scale it automatically sends the data to HA. Really convenient to use.

I have a Xiaomi scale connected through an esp Bluetooth proxy. Works pretty well.