I might have some experience you could find helpful , as I also live in a rural area, and I have a lot of day to day chores I do out in the yard... The first year I lived with my F7 SBT, I took him out with me wearing a leash and walking jacket while I did chores. He was an only kitten and I felt really bad to leave him in the house with access to just the enclosed porch while I worked, and he took full advantage of that and would meow pitifully if I tried to leave him behind. He is really well behaved on a leash, but it was like working with a toddler. He would get tangled often, get bored and want to be moved around, and any time I needed to do something involving 2 parts of the yard that were too far apart to have eyes on him, (often) it really slowed me down, and was extremely impractical. Cats that can go out on a leash are not at all like dogs! And then he got totally obsessed wanting to be out at least 8 hours a day... and even twice a day for an hour, soon after we came back in, he was watching me like a hawk in hopes of getting to go out again... I built a pretty deluxe series of enclosures ( deluxe from a cats point of view, but largely made from salvaged junk) and this did not help. I got him a feline buddy, which he seems to enjoy, but he still wanted out. The new F6 kitten who was normally contented would howl if I went out with the older cat, and having them both out there was like managing a 3 ring circus. Nothing else could be done...As a solution I ended up putting a cat proof fence around the main part of my yard and some of the wilder side areas. And this has worked... ( I notice the link Patti posted says fences won't keep a Savannah cat in, but cbain who has post here showing his enclosure has supplemented his fence with a small animal electric fence, running along the top and upper sides, and he has managed to keep his F2 and F3s in for years. I am using a similar system for my F7 and F6, and it seems to be working, though it has only been up a couple months.
I don't know why HB became so desperate to be out in the yard.... Maybe just his personality... my F6 seemed content with just the enclosures... But I was careful to only bring him out maybe once a month... just enough he wouldn't be totally disoriented if he ever accidentally got out. It maybe taking HB out into the yard every day, when he can see this area from the enclosures, meant he knew what he was missing , and seeing the places he loved, just out of reach really bothered him. I always carried him out the door in a carrier, but in retrospect, maybe we would not have had this problem if I carried him 1/4 mile down the road, before letting him out to enjoy himself. That way, he wouldn't be looking at where he loved to be all the time...? But on the other hand, once I accidentally left a basement door open, and because he was so familiar with the yard he just went to his favorite spot and thought he had gone for a walk without me. He is very familiar with the mile around our house, and could easily find his way home and would avoid the one road, if he ever accidentally got out. He probably would not have learned how to get home so well, if I carried him well away from our house before letting him walk. Another factor is there is a lot of rodents here, and as soon as he began catching them, in the house, in his enclosures, and while out on a leash, he began wanting out. So it maybe this experience woke up his hunting genes, and that is why he wanted out to do what cats do. But my F6 also likes to hunt, and now he has a choice, often chooses to do this in the enclosure rather than the yard... so I don't know... but I do think that having a highly social active indoor cat may be a lot easier if you are also an indoor person. If you live in a outdoor cat paradise, and want to leave your cat in an enclosure while you enjoy that cat paradise without your cat, there may be some challenges... Just my experience...!