Let me see if I understand your position correctly.
Unattached moveable personal property can not be claimed by the new owner of the house.....
Not exactly.
Since the new owner does not acquire ownership of the unattached, moveable, personal property in the house by buying the house, unless the property is specified as included in the purchase documents, the unattached, moveable, personal property is still owned by the seller of the house. There might, however, be ways in which the new owner of the house can claim ownership; and how that might be done will depend on state law. For example, the laws of a State might provide that if the finder of the lost property takes reasonable steps to find and notify the owner of that property that it's been found, and the owner then fails to claim it after some specified period of time, the finder becomes the owner.
Can you cite any law requiring a different conclusion? I have cited examples, viz., Georgia law, Kentucky law, and California law, each to the effect that someone who finds lost property doesn't automatically acquire ownership of that property.
...The previous owner of the house can claim the property after the sale of the home has been completed.
You have not answered my question as to how long the previous owner of the house has to reclaim their property. Is it months or even years?....
It will depend on state law and what steps the new owner of the house has taken to find and notify the owner of the personal property that it's been found.
...Since your position is claiming unattached property left behind in a house after the sale is theft then wouldn’t the statute of limitations apply?.....
Again, it will depend on state law and what steps the new owner of the house has taken to find and notify the owner of the personal property that it's been found.
....So the new owner of the home is required to save the property until statute of limitations runs out to avoid being charged with a crime.....
Again, it will depend on state law and what steps the new owner of the house has taken to find and notify the owner of the personal property that it's been found.
...It would also logically follow that since the new homeowner has to keep the property he is also required to keep it in the condition it was left in. Failure to do so would make him subject to a civil lawsuit for damages. So the homeowner is required to provide storage space even if requires renting a storage unit to do so.....
In general the new owner of the home, as a gratuitous bailee, would have only a very limited duty to preserve the personal property he fortuitously finds in his custody. And any possible liability can be limited by promptly taking reasonable steps, consistent with applicable state law, to notify the owner of the personal property that it's been found and will be disposed of if not claimed within some period of time.
If the personal property left behind is sufficient in size or volume to require any appreciable amount of storage space or renting a storage unity, it should be obvious immediately upon moving in. At that time it should be possible to promptly notify the seller of the house that his property has been found and will need to be removed.