Console popular nurse beset by rejection (8)

I believe the answer is:

nintendo

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'console' is the definition.
(games console)

'popular nurse beset by rejection' is the wordplay.
'popular' becomes 'in' ('in' can mean trendy).
'nurse' becomes 'tend' (tend can mean to nurse or look after).
'beset by' means one lot of letters goes inside another ('beset' could historically mean 'cover').
'rejection' becomes 'no' (I can't justify this - if you can you should give a lot more credence to this answer).
'in'+'tend'='intend'
'intend' going inside 'no' is 'NINTENDO'.

Can you help me to learn more?

(Other definitions for nintendo that I've seen before include "People behind Mario" , "Console producer" .)

I've seen this clue in The Independent.

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