BedJet Climate component not updating MQTT

I am configuring a node to connect to my BedJet V3 and publish updates to MQTT. I don’t want to use the native API as I have a specific algorithm in mind I want to implement via an MQTT client, maybe Node-RED.

The BedJet-related part of my YAML file looks like this:

bedjet:
  - id: bedjet_id
    ble_client_id: bedjet_ble_id

climate:
  - platform: bedjet
    bedjet_id: bedjet_id
    id: bedjet_climate_id
    heat_mode: extended
    temperature_source: outlet
    action_state_topic: "esphome/nott/bedjet/action"
    fan_mode_state_topic: "esphome/nott/bedjet/fan"
    mode_state_topic: "esphome/nott/bedjet/mode"
    preset_state_topic: "esphome/nott/bedjet/preset"

sensor:
  - platform: bedjet
    bedjet_id: bedjet_id
    outlet_temperature:
      name: "Outlet Temperature"
      state_topic: "esphome/nott/bedjet/outlet_temperature"
    ambient_temperature:
      name: "Ambient Temperature"
      state_topic: "esphome/nott/bedjet/ambient_temperature"

The problem I am having is that information from the sensors is published to MQTT, but absolutely nothing from the climate component.

As I understand it, that should happen “automagically” since the BedJet climate component inherits from the base Climate component.

Is there something missing from my configuration? Is there an issue with the base Climate component?