There is NO WAY Putin could have ordered such stupid and harmful act. As Machiavelli said: the skillful politician do not make OBVIOUS mistakes. (Non-obvious mistakes everybody can make).
It is quite likely that the Litvinenko death was an accident taking place among deranged people playing with dangerous toys. But if there was a real conspiracy, the most likely it was done in order to damage Russian government. The key could be the failed assassination of former prime minister Gaidar. Gaidar was the main proponent of 1990s free market reforms and his death could be used against Putin in a much more powerful way..
When Gaidar managed to survived, he did not want Western doctors to help him. He fled immediately to the Russian embassy and then flew to Russia. This strange event must have made him rethink his allegiances.
This is the point of view I am leaning towards as well. But then again there is always a first time for a slip-up even by a shrewd political animal.
And then there also is the slightest of possibilities that what appears obvious is right and true because many would claim it to be impossible and conveniently look in other directions anyway.