lord of the rings - Why did Saruman lose his power after he was banished from Orthanc?

I don't think the books ever went into much detail, but from this...

Book III, ch.10:

‘Be­hold, I am not Gan­dalf the Grey, whom you be­trayed. I am
Gan­dalf the White, who has re­turned from death. You have no colour
now, and I cast you from the order and from the Coun­cil.’

He raised his hand, and spoke slowly in a clear cold voice. ‘Saru­man,
your staff is bro­ken.’ There was a crack, and the staff split
asun­der in Saru­man’s hand, and the head of it fell down at
Gan­dalf’s feet. ‘Go!’ said Gan­dalf. With a cry Saru­man fell back
and crawled away.

..and this...

Book 6, Ch.8:

A sud­den light broke on Frodo. ‘Sharkey!’ he cried. Saru­man laughed.

‘So you have heard the name, have you? All my peo­ple used to call me
that in Isen­gard, I be­lieve. A sign of af­fec­tion, pos­si­bly.
But ev­i­dently you did not ex­pect to see me here.’

‘I did not,’ said
Frodo. ‘But I might have guessed. A lit­tle mis­chief in a mean way:
Gan­dalf warned me that you were still ca­pa­ble of it.’

‘Quite ca­pa­ble,’ said Saru­man, ‘and more than a lit­tle...

...it would seem that Gandalf removed Sarman's power. At least, Saruman's greatest capital-P Powers; Saruman still has some abilities, but nothing beyond the reach of any "ordinary" (i.e. non-Maiar) being in Middle-earth. Could Saruman have redeemed himself? Maybe,...

Book III, Ch.10:

‘He will have guessed, surely?[re. Treebeard]’ said Merry, ‘Were they
likely to end any other way?’

‘Not likely,’ an­swered Gan­dalf, ‘though they came to the bal­ance of
a hair. But I had rea­sons for try­ing; some mer­ci­ful and some less
so. First Saru­man was shown that the power of his voice was
wan­ing...Then I gave him a last choice and a fair one: to re­nounce
both Mor­dor and his pri­vate schemes, and make amends by help­ing us
in our need. He knows our need, none bet­ter. Great ser­vice he could
have ren­dered. But he has cho­sen to with­hold it, and keep the power
of Or­thanc...He lives now in ter­ror of the shadow of Mor­dor, and
yet he still dreams of rid­ing the storm. Un­happy fool! He will be
de­voured, if the power of the East stretches out its arms to
Isen­gard.

...but Saruman was given a last choice, and he still chose "the dark side".

Did Gandalf remove Saruman's power himself, or was it a Higher Power (the Valar or Ilúvatar themself)? I think it depends on how you want to look at it. An executioner acts on the order of the Monarch/Judge/whatever, with the "Power" of the throne/gang/society/legal system behind them, but it's still the executioner that pulls the lever/chops the head/pushes the syringe. If Saruman had shown any redeeming qualities, Gandalf could have chosen to give him a second chance. But Saruman refused the offer, so Gandalf chose to break him. In the end, it's always the choice of the person with their finger on the trigger. Which is a very Roman Catholic point-of-view, but as Tolkien was a Catholic I think that's how it was meant to be read.