Hello Roey,
Quirk's "A Student's Grammar" tells that you can do the way you like more
. That is,you may " ..examine it as a elliptical version of clause coordination,noting what elements are ellipted or you may examine the construction as you see it i.e noting what elements are present.In your example you can view the sentence as two independent clauses connected with the coordinator "but",implying the subject in the second sentence:
She was tired
but
(She) didn't go to sleep
Or you may examine it as a one single clause with coordinated predicate:
She was tired but didn't go to sleep.
Without giving reasons for it , the preference is given to analysing it as you see it rather than as elliptical,specially in the case of the simple coordination.
Note also that you can leave out the subject only if you join coordinated ideas but not if one idea is subordinated to the other,like:
*She was tired because didn't go to sleep.
Best regards
