Unfortunately there's not really a simple answer to your question as fabric huts and toys can be safe or they can be deadly. There are stories all over the internet of birds who have died from those huts but there are also stories of birds using them safely for their lifespan.
On the "deadly" side there are birds chewing on the fabric and then becoming entangled in the loose strings and breaking their necks or choking to death. Sometimes they don't die but they lose toes or wings. There are also stories of birds ingesting pieces of fabric which is usually deadly.
In fact one of the members here who has a green cheek conure recently took her bird to the vet and the vet discovered a blockage of fabric (from a rope perch) in its digestive system and said the bird was lucky to be alive because usually the only time you see that is in a necropsy once the bird has died from it.
On the "safe" side some birds have huts and don't ever chew on them or become entangled. My green cheek conure sleeps in a Happy Hut and he had one in his first home and possibly even the pet store. He is 7 now and I've never had a problem with him chewing on the fabric or doing anything other than just sleeping in it. However, I check it every single day for bald spots, loose fabric, strings, etc. because the reality is fabric huts will never be 100% safe and if I ever noticed signs he was chewing on it, I would remove it. Even still I can't be sure an accident won't happen but, as I said, he only seems to sleep in it and other than that does not spend time in it and as I check its condition daily I don't really worry about it too much. Hopefully it will not come back to haunt me in the future.
A possible half-way solution is an "avian hut" which is kind of like a little tent that hangs over a regular perch. It's still made of fabric but since your bird is perched on an actual perch instead of on fabric there is less of a risk of entanglement of the nails, toes, etc. Here are links to two: Avian Haven Bird Tent