When I was working away from home for 8 yrs not so long ago. "Retired last week" I was worried that thieves would break into my house when away for weeks on end.
I started out with a Canary in my living room then added two more, one on my back patio and one in my boat house.
You don't have to worry about someone stealing the camera because before they can spot it, the footage has already been uploaded to the Cloud for your viewing from any place at anytime.
I removed the boat house cam and replaced it with a Ring floodlight cam. I had video of people coming up to my boathouse at night, getting out on the boathouse and fishing around my underwater green light.
I realized that these people didn't know they were being recorded so that's the reason for the floodlight cam. I also posted signs on three sides of my boat house stating "WARNING: 24 HR VIDEO SURVEILLANCE VIOLATORS WILL BE PROSECUTED" I've had no problem since I did that. Actually hardly anyone even fishes around my boathouse anymore which is legal.
I moved the Canary camera from the boat house to one side of my house then, I added a Ring Doorbell cam. I've got five cameras up and operating now and I catch some weird stuff.
All of my cameras record voice and I can speak through any of my cameras too. The one in my living room and the one at the boat house have audible alarms if I choose to set them off.
If anyone comes near this place, I have a video of them. The drawbacks are the cameras are sensitive. I have videos of spiders, bugs flying past the camera, etc.
Oh, if I have a power outage, all of my cameras are on battery backup except the floodlight cam at the boathouse. My WiFi is on battery backup as well.
Now getting back to your idea of a game cam. I had one set up at my RV in Houston. It was camo and I mounted it about 7' up in a tree looking down at an angle. It lasted until I left and worked well but all you get is pics unless you set it up for video. The problem with just pics is the person may have their head turned when it takes the pic.
If you plan to check it often, Vid will work best but it will use a lot more memory and power. You can now get game cams that upload to the cloud but they could be pricey plus you would need some sort of wireless plan and signal.