The switch you see is from the fritzbox integration. (you see 2 integrations on the left for the device). It shows if the device is allowed internet access, and this particular switch doesn’t allow changing from HA. It is off for me too, and the proxy works just fine. (It should have been a binary sensor, not a switch in the first place).
That is not to say you may not have a problem if the esp disconnects, it just isn’t related to the switch, nor can you set it other than from the fritzbox by assigning it a profile. In general it is good practice that iot devices have no internet access if they do not need it, and this one does not.
The safe mode boot entity is unknown for me too, I think buttons are always in that state? What makes you think the proxy is not connected? The router says it is online (home), according to the other fritzbox added entity. My working proxys look the same.
So the question goes back to: what bluetooth device where you hoping to discover? Mobile phones you can probably add if they have a fixed mac. They are not autodiscovered unless you autodiscover any ble device (not recommended, it will explode with entities). You need to add them to known_devices.yaml if you want to track them:
ble_track_mydevice:
mac: BLE_40:92:28:59:E5:5A
name: 'MyDevice'
icon: mdi:bluetooth
track: true
BLE thermometers could be autodetected, depends on which one. I have Inkbird, those were found.