Why would it be an advantage to have terminal dates?
Because at that point some party has to deliver a real asset. The market must converge to a true price since a physical commodity is changing hands.
Are you sure you are getting oil or gold when you are betting on it with a centralized system?

I think changing physical things is not necessary and not done in existing betting markets. But what they have is an index that represents the market where the real physical assets are exchanged.
Correct me if I am wrong: In a system without an index given by a central authority you rely on everyone's assumption that everyone will follow the price of the underlying asset (must be precisely defined) so that it makes sense to bet on where the asset's price will be in the near future. Near future here is defined for short positions as up to the point where their asset looses 50% and for long positions there is no termination (I dont see a problem in the latter).
There are two assumptions/implications for this to work:
(1) The asset is exactly defined (for example 1 barrel of WTI brand; or whatever is traded physically).
(2) It is habitual behavior where the shared understanding to follow the price of the underlying asset is reinforced the more people do it. And you win when you predict the future price that deviates from the current price right as long as no short squeeze takes place (when you are short and your asset looses 50%) -> What is therefore needed or at least helpful is reputation which creates mass / herd behavior. If a big bank puts out such a system the mass would easily rely on that everyone is following the underlying asset price
*I used the word "underlying asset price" as the price that is actually paid for a good when it is physically exchanged.