I don't find care or ownership of parrots to be isolating. My GCCs have a ginormous cage with lots of toys. They spend their day in it. When I'm home, I'll have them out with me while I'm puttering around the house. If I have guests over, it depends on the guest and what's going on at my place - if the birds can be out or the guests want to play with them, they're out. If we're doing something that'd be unsafe for the birds or they aren't comfortable with the birds or the birds are scared of them, the birds are in their cage (but in the same area, so able to spy on us). Hasn't really been any issues.

Here's the thing - I think you need to reframe your thinking about cages. Your birds are already locked in a cage 24/7. That cage is your house. It doesn't really matter that it's a big space, it's still a cage.

Here's the other thing - the cage you're housing your birds in is probably dangerous to their health. Do you use electronics? Leave things plugged in? Cords can be chewed through. A TV can be knocked over or the screen broken (seen pictures of it on here, even!). Blanket or pillows can be torn apart, feet can be caught in threads, birds can die before you get home. If they're free range 24/7, what's to stop them from getting slammed in a door or flying out when you're entering/exiting your house? Accidents like that are posted on here all the time.

The cage you're housing your birds in is also dirty. Or at least, that's what I get from your cleaning up after them but not always able to get everything. That's not good for your health or their health.

I'd seriously consider setting aside a bird-proofed room or aviary for them (with easy to clean features, like vinyl flooring) if you want them to have more space when you're not home. Then when you are home, they can be with you when you're supervising them. If you can't set aside a room, get a big cage and deck it out. There are all sorts of options out there, from flight cages (or double flight cages) to walk-in style aviaries that are even larger.