I was able to make it work. Seems ESPHomeFlasher (i prefer to use as it is very stable and using for years now) has a problem with ESP32-C3 as it writes from 0x00001000 (not sure how it handles boot loader).
Using browser based HomeAssistant ESPHome built in tool it works, it writes from 0x000000 and creates proper bootloader.
Config snippet in case someone else faces install or build problems (BLEProxy requires CONFIG_BT_BLE_42_FEATURES_SUPPORTED) :
esphome:
name: esp32ble-proxy-131
platformio_options:
board_build.flash_mode: dio
esp32:
board: esp32-c3-devkitm-1
framework:
type: esp-idf
sdkconfig_options:
CONFIG_BT_BLE_42_FEATURES_SUPPORTED: y
variant: ESP32C3
[12:32:26][C][esp32_ble_tracker:793]: BLE Tracker:
[12:32:26][C][esp32_ble_tracker:794]: Scan Duration: 300 s
[12:32:26][C][esp32_ble_tracker:795]: Scan Interval: 320.0 ms
[12:32:26][C][esp32_ble_tracker:796]: Scan Window: 30.0 ms
[12:32:26][C][esp32_ble_tracker:797]: Scan Type: ACTIVE
[12:32:26][C][esp32_ble_tracker:798]: Continuous Scanning: True
[12:32:26][C][bluetooth_proxy:167]: Bluetooth Proxy: