Does the word “exempt” always work in favour of the exempted party?

Exempt may be the wrong word. It's usually exclusionary in a positive sense: John is exempt from prosecution because he has agreed to testify against the guilty parties. His class was drafted, but John was exempted due to health issues.

"John is exempted the right of free accommodation," is like saying, "Everybody but John gets free room and board."

John is extended (offered) the right of free accommodation makes more sense.

Perhaps "extended"; not "exempted", is what you are looking for.

ex·empt
iɡˈzem(p)t/
verb
past tense: exempted; past participle: exempted
free (a person or organization) from an obligation or liability imposed on others.
"they were exempted from paying the tax"
synonyms: free from, not liable to, not subject to, exempted from, excepted from, excused of/from, absolved of More
https://www.google.com/search?q=exempted&rlz=1C1GGGE_enUS517US518&oq=exempted&aqs=chrome..69i57&sourceid=chrome&es_sm=122&ie=UTF-8

Hope this helps.

Yes, "exempt" is similar to "saved from." "Excluded" might be what you want: John is excluded from the right of free accommodation. Or, more simply: John is denied the right of free accommodation.