Heartless institution got rid of those who were unwilling? (10)

I believe the answer is:

indisposed

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'unwilling?' is the definition.
(I've seen this in another clue)

'heartless institution got rid of' is the wordplay.
'heartless' says to hollow out the word (remove centre letters).
'got rid of' becomes 'disposed' (I've seen this before).
'institution' with its centre removed is 'in'.
'in'+'disposed'='INDISPOSED'

'those who were' is the link.
This may not be right. Some or all of it may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for indisposed that I've seen before include "Slightly unwell" , "Out of sorts" , "Poorly" .)

I've seen this clue in The Irish Times.

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