Here's a good site:
Would "'εληξε" be used in everyday Greek? Can we also say: "έχει τελειώσει" η "τελείωσε".
Yes, indeed. You are right. Your proposals are more common in everyday Greek.
I guess they haven't added it to the database. You should report it.
I probably did not include 'έχει τελειώσει' because I would expect to have the sentence : '.... has been over.' Am I wrong?
'έληξε' is used by me in everyday life as in 'Δε θέλεις να κάνεις τα μαθήματά σου; Τότε δε θα βγεις έξω. Το θέμα έληξε.' (=You don't want to do your homework? Then you won't go out. Subject closed.) You can imagine who I'm talking to... and occasionally I am considered ancient by them :D