The ac only powers a wall wart charger type system for the 12v batteries.
The ac has nothing to do with the gate opener other than to charge a battery.
The ac does not over ride a dead battery. The battery is the only thing that opens the gate.
If you don't install the batteries, the gate won't work. If your batteries go bad, the gate won't work.
The gate runs on the batteries only.
The gates run perfectly on a $24 solar panel.
The batteries commonly used, the ones that fit within the opener control housing, will open and close the gate (if a single) about 50 times without charging.
If that is not enough times without daylight, a cheap car battery is just a jumper wire away from allowing it to open 100 times.
Your chosen opener is 24vdc motored.
Double check me on all of this.
If it will run on battery back-up, that means it doesn't need ac power.
Running the ac literally does nothing except cost you money.
(I worked on a liftmaster just yesterday. Some turd pushed it open by hand and I had to do a reset so it could find its limits.)
And
400ft is a long way for a liftmaster remote in many landscapes.
I spliced a super long distance $15 amazon remote control right into the 'emergency open' line switch because some people are just too stupid, too unknown, or too untrustworthy to either be given the code or work the keypad.
And
https://www.amazon.com/gp/aw/d/B07GFDQ467?psc=1&ref=ppx_pop_mob_b_asin_image
This spliced into that same line allows Alexa and Google voice or app control.
Me: echo, open the gate.
Ecbo: 'OK. What fresh hell will this become?'
Or
Me:echo, open gate
Exho:'ok, send in the clowns. The damned clowns'