I had a similar situation with my shutter blinds. The company wanted €500 to install a "smart hub" to enable voice control instead of their little remote. To hell with that, I ordered a couple of servos and an esp8266 and 3d printed some little fingers that would press the buttons on the remote and added the whole thing to HomeAssistant via ESPHome. Cost me less than €30.
I disassembled one of our gate opener's remotes, and hooked the button contacts to Raspberry Pi Zero's GPIO pins. Pi runs a tiny go webserver and Cloudflare tunnel. I hit an obscure HTTPS URL (via button on a phone's homescreen), gate opens.
Basically, they had your setup exactly, but they leave the tab open in Safari and whenever they open Safari to browse the internet, the garage door opens.