rtl_433-hass-addons VS bluetooth-proxies

Another option is an RTL-SDR antenna like this, paired with the RTL_433 add-on in HA. Then you can use any supported 433MHz sensors like this and this. 433MHz isn't a mesh network like Zigbee, but it does have longer range, doesn't require any pairing, and the sensors send updates every 16s, and take standard AAA batteries.

An RTL-SDR antenna like this attached to your HA server running this add-on, and a few 433MHz RF temp/humidity sensors like this and this.

Run one of the rtl_433 variants on your pi to capture the data, and Grafana/InfluxDB to handle the data once on your pi. (I use this version, but it was made specifically to integrate with HA.)

There's this one, that auto-discovers things. If you live in the middle of no-where that might be OK. Personally, there's like thousands of 433MHz around me (mostly car tire pressure sensors), so having HA ingest all of those would be a nightmare.

If you're already using Home Assistant, you can install ESPHome on any ESP32-based device and use it as a Bluetooth Proxy hub. I already had some smartplugs and bulbs flashed with ESPHome, so it was as simple as adding a couple more lines of code to get a few BLE Proxy hubs, with no additional hardware required for the hub.

Home Assistant and Bluetooth Proxy is probably your best bet: https://esphome.github.io/bluetooth-proxies/

where can I find the yaml configuration file? I flashed the esp32 directly with the bluetooth proxy wizard, hence I never touched the yaml file myself. Is it stored somewhere in HA?

Yep, these are the way to go. Small and with a case, so ready to put around the house.

Using just your browser you can transform these into Bluetooth proxies for Home Assistant with https://esphome.github.io/bluetooth-proxies/

I think the sequence when using the web installer from here is that it first installed a basic EXPHome WiFi connector and once done, supposed to autmomagically install the BT Proxy software but it fails. It gives you the chance to rename it to something more appropriate but then you get some other 'web' ESP device pop up?

ESPHome uses a yaml file as input from which to generate the binary firmware file for your ESP. Flashing via their github as you did means it will use one of these predefined yaml files. Because you chose generic esp, it will be using this one. As you can see it already has the api: line in there (line 21).

with framework set to esp-idf (as per https://github.com/esphome/bluetooth-proxies/blob/main/gl-s10.yaml), the device logs stop at "Starting scheduler on APP CPU"