The associated reviews and product availability on Amazon are not encouraging. Having said that, the concept is similar to a tiny garage door or roller shutter. There has been a lot of discussion on various aspects of a chicken coop here in these forums that may give you ideas. Just use ‘chicken coop’ as your search criteria.
Yes, you can take the simple route and have a device that presses the remote buttons for you, or tap into the sensors and driver, and essentially replicate the existing controls with a replacement add-on. Your sensors will probably be light level, door position, limit switch for obstructions, voltage of power supply, and maybe a proximity sensor tuned to chickens. Your control will be signals for the door to open and close. Whether you just tap into the controller to do that or drive the motor directly is up to you. You will need some form of keeping correct time, either via network or onboard your automation device. Visual chicken facial recognition to prevent foxes entering would probably be gilding the lily, but you may want a two way video/audio feed to greet them individually like in the videos!!! Depends on how much they are pets or just egg producing livestock is going to determine how much time and effort you spend on this.
If you want specific help on how to add your sensors to ESPHome, some photos of the various boards would help. Even both sides of your remote control, detailed enough so we can read the chip numbers and board markings would be the first thing to do. It looks to be only very short range of operation. Any FCC ID markings on equipment or the manual? This may assist in identifying the remote technology being used.
Photo of the power supply, including the ratings would also assist, as now you will have the added requirement to power your remotely controlled automation, which will shorten the time the battery retains adequate power.
Distance is also a factor. If it is too far away, you may need LoRa rather than ZigBee or WiFi to establish reliable two way communications.
How far away is the closest power from the unit? For the one that is not close, you may have to install another bigger solar powered supply or an addon just for the automation to keep everything running.
Don’t be overwhelmed. An ESP32 running ESPHome with a cheap two port relay board, one channel for up, and another for down may do the trick, or just a onboard radio to emulate the remote control signals to the door will work. If you want fancy, then we can also rise to the challenge…
Reverse engineering how it all works is the challenge. Post detailed photos of the remote controller board, both sides, to start with, along with any labels and FCC markings for all parts. Let us know how confident you are with wiring and hooking up added devices to ESPHome and HomeAssistant.