Schlage LE could work, though made for offices & multi-tenant. Expensive too

It's off topic, so I don't know how much I should get into my phone's lobotomy. But you asked, so I'll try to explain as briefly as I can, apologize in advance for the OT post, and hope that people who aren't interested will skip over this rather than getting upset with me. The why: What I said above. I just don't want my phone sending my personal data to some computer over which I have no control, for purposes that I don't consider useful. I also pay for every megabyte (prepaid service), and I'm cheap. The how: The right way to start would probably have been to flash an open source ROM and not load the optional Google apps. That was more than I wanted to get into. So, stock ROM, no Google account, sync off, GPS/data/wifi/bt off when I don't need them. Nothing in the cloud; contacts, calendar, email, and maps all on the phone. I prefer a dedicated GPS receiver, and use Osmand (Open Source Maps Android) for directions only when lost without one. Play Store, Google Services, search, weather, voice recognition, and many other background apps and services are disabled. Firefox Mobile and websites replace some common apps. No social media. Apps are mostly open source, all sideloaded. I even have an open source Youtube app. Updates are manual on my schedule. It's smarter than a dumbphone, but a little dumber than a normal smartphone, I don't miss most of the surgically removed features, because I use a real computer for real computer tasks, such as this one, and only use the phone as a computer when it's the only device available. Bonuses: Despite being over 7 years old, the phone is fast. The battery typically lasts 5-8 days. Service cost is usually around $5-10 per month. Probably most people wouldn't want a phone like mine, but it does exactly what I need and want it to do, and no more. Your mileage may vary.

Aside from the issue of whether a smart lock exists for your type of door, any door that is "a bit hard to open and close" is not a good candidate for a smart lock as the bolt needs to be able to travel freely without issues of friction or alignment with the strike plate.

@Jae Bourg So I was imagining a normal WiFi or other wireless technology smart lock such as this one by Schlage: https://www.amazon.com/Schlage-Deadbolt-BE489WB-CEN-622/dp/B07HXMLCBT To make that work, you need to rebuild the gate to use a fence post frame, so that normal home door lock designs can be used on the door fence post up against the fence post of the frame. That will cost more than $500, unfortunately. The nice feature that this enables is the ability to set temporary access codes, open the gate remotely, and set time zones. For example, I might enable a gardener code to work Saturday through Monday, from 9 am to 2 pm. That way if some employee of the gardener ends up being a bad guy and shows up at 3 am, the code is not going to work.