RP2040: MQTT client support for Raspberry Pi Pico W

**Describe the problem you have/What new integration you would like**

I would like to see MQTT client support for ESPHome on the Raspberry Pi Pico W platform.

The existing MQTT client only supports the ESP8266 and ESP32 platforms.

- https://github.com/OttoWinter/async-mqtt-client

Previously, attempts to build ESPHome with MQTT for Raspberry Pi Pico W failed with compilation errors.

```
# file: picodebug.yaml
esphome:
name: picodebug
build_path: /tmp/.esphome

rp2040:
board: rpipicow
framework:
platform_version: https://github.com/maxgerhardt/platform-raspberrypi.git

logger:
api:
password: ""
ota:
password: ""
wifi:
ssid: !secret wifi_ssid
password: !secret wifi_password

mqtt:
broker: 192.0.2.1
```

- the JSON component wants a free_heap number but it's hardware specific and only available for ESP8266/ESP32
- the MQTT library similarly only support ESP8266/ESP32

Example:

```
Compiling /tmp/.esphome/.pioenvs/picodebug/src/esphome/components/json/json_util.cpp.o
src/esphome/components/json/json_util.cpp: In function 'std::string esphome::json::build_json(const json_build_t&)':
src/esphome/components/json/json_util.cpp:29:34: error: 'free_heap' was not declared in this scope
29 | size_t request_size = std::min(free_heap, (size_t) 512);
| ^~~~~~
```
- see https://github.com/esphome/esphome/blob/dev/esphome/components/json/json_util.cpp#L19-L29

...and:

```
Compiling /tmp/.esphome/.pioenvs/picodebug/src/esphome/components/mqtt/custom_mqtt_device.cpp.o
In file included from /tmp/.esphome/.piolibdeps/picodebug/AsyncMqttClient-esphome/src/AsyncMqttClient.h:4,
from src/esphome/components/mqtt/mqtt_backend_arduino.h:6,
from src/esphome/components/mqtt/mqtt_client.h:15,
from src/esphome/components/mqtt/custom_mqtt_device.h:7,
from src/esphome/components/mqtt/custom_mqtt_device.cpp:1:
/tmp/.esphome/.piolibdeps/picodebug/AsyncMqttClient-esphome/src/AsyncMqttClient.hpp:14:2: error: #error Platform not supported
14 | #error Platform not supported
```

- see https://github.com/OttoWinter/async-mqtt-client/blob/master/src/AsyncMqttClient.hpp#L8-L15

The MQTT component has since been disabled for the RP2040 platform:

- https://github.com/esphome/esphome/pull/4025

**Please describe your use case for this integration and alternatives you've tried:**

I would like to send sensor data over MQTT for recording outside of Home Assistant (HA) as well as do signalling with other services available only over MQTT.

**Additional context**

It might be worth considering that there are open discussions on the topic of MQTT clients:

- https://github.com/esphome/feature-requests/issues/811

NOTE: Signal your interest in this feature with the "thumbs up" on this post. No need to spam the issue thread with "me too" comments (they cause unnecessary email notifications).