My birds live in an outdoor aviary so they are cage-free and "unsupervised" for most of the day, but since they are in a space that has been carefully designed for bird safety, I am not that worried about them. I check on them multiple times each day, but I don't spend every available second watching them ... although I do love hanging out in the aviary, just chilling.
Leaving birds unsupervised in a human household is much more risky due to the wide range of perfectly normal hazards that exist in the average household - electrical wires, toilet bowls, tight spaces, poisonous substances, open doors/windows, etc. If you are watching your bird, you can rescue it, if it falls into an open box and pulls a blanket down on top of it or lands in a pot of water on the stove or starts chewing on a box of dish soap. But if you leave the room, your bird might be injured or dead by the time you return. A lot can happen, even in a few minutes, let alone hours, of unsupervised time in an open room. And if a door or window is left open, the bird can be gone in mere moments.
On the other hand, if they are in an indoor aviary or carefully designed bird room, they might be just as safe as they would be in a small cage. If the space is 100% bird safe and all hazards have been carefully removed, I don't see any problem with leaving the bird unattended. The tricky part is ensuring that the entire space is really safe AND that the bird can't exit the safe space unintentionally.
I'd also mention that some birds are more trustworthy than others. A fiesty little parrotlet who is constantly getting into trouble and discovering new things to chew is a poor candidate for unsupervised time, while a calm older parrot who spends most of her time sitting on a play tree and looking out windows is a much safer bet.
Ideally, your home should be completely bird safe, but realistically, that is rarely 100% true. Any amount of unattended time is taking the risk that something might happen to your bird while you are too far away to help them. But if you take the time to do proper risk assessment and risk management, I do think it is possible to lower that risk to an acceptable level, especially in your own home where you can exert a lot of control over the space.
Personally, I am a big advocate for bird rooms and both indoor and outdoor aviary setups. I think that we should strive to provide our birds with as much freedom as we safely can for their physical and emotional wellbeing. But not everyone is in a position to create a dedicated bird space. And if your birds are in a room that was designed for people, not birds, it is important to be extra vigilant to protect your birds from the hidden dangers that are likely present. A big part of that vigilance involves not leaving your birds unsupervised in a potentially hazardous location.
Better safe than sorry!
