Photographer here. You're pretty much spot on. The reason for why this happens has got to do with three reasons, two of which are pretty hard to overcome:

1. Wide angle lens, short focal length. You can't fit a long focal length lens on a phone, because apparently no one wants a super thick phone these days. To get photos of the Moon with any reasonable detail you need a pretty decent telephoto lens (I get fairly good results with my 200 mm prime and so-so results with my cheapo 300 mm zoom)

2. Resolution. So not only can you only fit a small lens in the phone, you also just can't physically fit a large sensor behind it either. You have a tiny lens which passes through a tiny image on a tiny sensor, so phone makers have long since hit the physical limits.

3. Control software. Photographing the Moon is a special occasion as far camera automatics is concerned. The Moon is a bright object. It reflects daylight, dammit. Robot brain cannot comprehend this. In the nighttime, camera automatics scream "Aah! High ISO! Long exposure! Wide aperture!" You need to be able to tell the camera you really want daylight ISO and daylight exposure time and daylight aperture. (usual rule of thumb: ISO 100, 1/100 seconds, f/11) Now, the software in phones tries to usually approach this by letting you specify scenarios, but even the vague "night mode" is hit and miss for me sometimes. Fortunately this is something that is usually easy to rectify, because it's a software issue. (Open Camera for Android is pretty sweet, gives you full manual mode.)

It's the human eye and brain that increase the subjective size of the moon.

To test this, turn away from the moon and hold your thumb and index finger at arm's length in front of your face and apart just enough to touch the edges of the moon. Without bringing your thumb and index finger closer together, and still keeping them at arm's turn back towards the moon and see if you got the size right. You'll find that the size of the moon is closer to what your camera displayed.

I've heard that neweriPhone camera "AI" adds made-up details to moon pictures, but I don't think they increase the subjective size.

Well partly, but part of it is just no one has a very good mental idea of how big the moon is (if you hold out your hand at arm's length you should be able to cover it with the width of your thumb). When you see it pretty close to the horizon with actual context it looks a lot bigger than it really is. But then when you take a picture of it all that context goes away because you zoom in on the moon.