Without this legal process numerous homes and properties would remain abandoned forever with no clear way for someone to claim possession if the former owner has no heirs.

We always pro-rate rent to the last day of possession, without sweating over which specific day they will be out (as long as we did get the minimum required written notice (text messages are not acceptable, and likely not "sufficient notice" per local law) of their expected move date, and their existing term has been completed).

Tenants usually are living in the property in a way that their possessions and cleanliness affect the average buyer’s impression of the home (for example, misc dishes in the kitchen, pizza boxes, random clothes and boxes on the floor).

I would offer to be helpful and if they have a lawyer maybe talk to him about details of access and possession and dates (more neutral and less emotion there).

They are asking me to allow entry into house to take possession of deceased belongings as "spouse never bought anything in the house" and "tenant didn't have will".

After various delays, we finally took possession of the unit. 

Everyone should have a clear, written policy regarding tenants that fail to take possession.

If they do get a contract they’ve often analyzed the value of the property incorrectly, priced repairs and improvements wildly inaccurately, and do not possess the ability to shepherd a deal through the inevitable problems that pop up during the contract period.6.

Honestly, general advice here is to take possession vacant unless you are able/knowledgable in handling getting the tenants out somehow, legally.I've done estoppels before. 

We took back possession of the unit, not too bad of shape.